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William B. Wetherell
Optical Research Associates
Framingham , Massachusetts
F, F 9
h , h9
l , l9
P , P9
tan a
x, y, z
back focal length
pupil diameter
eye relief common pupil position
eye relief keplerian
exit pupil; eye space
focal points
front focal length
object and image heights
object and image distances
angular magnification
linear, lateral magnification
refractive index
object relief
entrance pupil; object space
principal points
total length
cartesian coordinates
axial separation
If collimated (parallel) light rays from an infinitely distant point source fall incident on the
input end of a lens system, rays exiting from the output end will show one of three
characteristics: (1) they will converge to a real point focus outside the lens system, (2) they
will appear to diverge from a virtual point focus within the lens system, or (3) they will
emerge as collimated rays that may differ in some characteristics from the incident
collimated rays. In cases 1 and 2, the paraxial imaging properties of the lens system can be
modeled accurately by a characteristic focal length and a set of fixed principal surfaces.
Such lens systems might be called focusing or focal lenses, but are usually referred to
simply as lenses. In case 3, a single finite focal length cannot model the paraxial
characteristics of the lens system; in effect, the focal length is infinite, with the output focal
point an infinite distance behind the lens, and the associated principal surface an infinite
distance in front of the lens. Such lens systems are referred to as afocal , or without focal
length. They will be called afocal lenses here, following the common practice of using
‘‘lens’’ to refer to both single element and multielement lens systems. They are the topic of
this chapter.
The first afocal lens was the galilean telescope (to be described later), a visual telescope
made famous by Galileo’s astronomical observations. It is now believed to have been
invented by Hans Lipperhey in 1608.1 Afocal lenses are usually thought of in the context of
viewing instruments or attachments to change the effective focal length of focusing lenses,
whose outputs are always collimated. In fact, afocal lenses can form real images of real
objects. A more useful distinction between focusing and afocal lenses concerns which
optical parameters are fixed, and which can vary in use. Focusing lenses have a fixed, finite
focal length, can produce real images for a wide range of object distances, and have a
linear magnification which varies with object distance. Afocal lenses have a fixed
magnification which is independent of object distance, and the range of object distances
yielding real images is severely restricted.
This chapter is divided into six sections, including this introduction. The second section
reviews the Gaussian (paraxial) image forming characteristics of afocal lenses and
compares them to the properties of focusing lenses. The importance of the optical
invariant in designing afocal lenses is discussed. The third section reviews the keplerian
telescope and its descendants, including both infinite conjugate and finite conjugate
variants. The fourth section discusses the galilean telescope and its descendants. Thin-lens
models are used in the third and fourth sections to define imaging characteristics and
design principles for afocal lenses. The fifth section reviews relay trains and periscopes.
The final section reviews reflecting and catadioptric afocal lenses.
This chapter is based on an earlier article by Wetherell.2 That article contains
derivations of many of the equations appearing here, as well as a more extensive list of
patents illustrating different types of afocal lens systems.
Afocal lenses differ from focusing lenses in ways that are not always obvious. It is useful to
review the basic image-forming characteristics of focusing lenses before defining the
characteristics unique to afocal lenses.
Focusing Lenses
In this chapter, all lens elements are assumed to be immersed in air, so that object space
and image space have the same index of refraction. Points in object space and image space
are represented by two rectangular coordinate systems (x , y , z ) and (x 9 , y 9 , z 9) , with the
prime indicating image space. The z - and z 9-axes form a common line in space, the optical
axis of the system. It is assumed, unless noted otherwise, that all lens elements are
rotationally symmetric with respect to the optical axis. Under these conditions, the imaging
geometry of a focusing lens can be defined in terms of two principal points P and P 9 , two
focal points F and F 9 , and a single characteristic focal length f , as shown in Fig. 1. P , P 9 , F ,
and F 9 all lie on the optical axis.
FIGURE 1 Imaging geometry of focusing lenses.
The focal points F and F 9 , will be the origins for the coordinate systems (x , y , z ) and
(x 9 , y 9 , z 9). If the origins are at P and P 9 , the coordinates will be given as (x , y , s ) and
(x 9 , y 9 , s 9) , where s 5 z 2 f and s 9 5 z 9 1 f. Normal right-hand sign conventions are used for
each set of coordinates, and light travels along the z -axis from negative z toward positive
z 9 , unless the optical system has internal mirrors. Figure 1a illustrates the terminology for
finite conjugate objects.
Object points and image points are assumed to lie in planes normal to the optical axis,
for paraxial computations. Object distance is specified by the axial distance to the object
surface, z or s , and image distance by z 9 or s 9. The two most commonly used equations
relating image distance to object distance are
1 1 1
2 5
s9 s f
zz 9 5 2f 2
For infinitely distant object points, z 9 5 0 and s 9 5 f , and the corresponding image points
will lie in the focal plane at F 9.
To determine the actual distance from object plane to image plane, it is necessary to
know the distance sp between P and P 9. The value of sp is a constant specific to each real
lens system, and may be either positive [moving object and image further apart than
predicted by Eqs. (1) or (2)] or negative (moving them closer together).
For rotationally symmetric systems, off-axis object and image coordinates can be
expressed by the object height h and image height h 9 , where h 2 5 x 2 1 y 2 and h 92 5 x 92 1 y 92 .
Object height and image height are related by the linear magnification m , where
h9 s9 z9 1 f
5 5
h s z 2f
Since the product zz 9 is a constant, Eq. (3) implies that magnification varies with object
The principal surfaces of a focusing lens intersect the optical axis at the principal points
P and P 9. In paraxial analysis, the principal surfaces are planes normal to the optical axis;
for real lenses, they may be curved. The principal surfaces are conjugate image surfaces for
which m 5 11.0. This property makes the raytrace construction shown in Fig. 1a possible,
since a ray traveling parallel to the optical axis in either object or image space must
intersect the focal point in the conjugate space, and must also intersect both principal
surfaces at the same height.
In real lenses, the object and image surfaces may be tilted or curved. Planes normal to
the optical axis are still used to define object and image positions for off-axis object points,
and to compute magnification. For tilted object surfaces, the properties of the principal
surfaces can be used to relate object surface and image surface tilt angles, as shown in Fig.
1b. Tilted object and image planes intersect the optical axis and the two principal planes.
The tilt angles with respect to the optical axis, u and u 9 , are defined by meridional rays
lying in the two surfaces. The points at which conjugate tilted planes intercept the optical
axis are defined by sa and s a9 , given by Eq. (1). Both object and image planes must intersect
their respective principal surfaces at the same height y , where y 5 sa tan u 5 s 9a tan u 9. It
follows that
tan u 9 sa
5 5
tan u s a9 ma
The geometry of Fig. 1b is known as the Scheimpflug condition , and Eq. (4) is the
Scheimpflug rule , relating image to object tilt. The magnification ma applies only to the
axial image.
The height off axis of an infinitely distant object is defined by the principal ray angle up
measured from F or P , as shown in Fig. 1c. In this case, the image height is
h 9 5 f tan up
A focusing lens which obeys Eq. (5) for all values of up within a specified range is said to
be distortion -free : if the object is a set of equally spaced parallel lines lying in an object
plane perpendicular to the optical axis, it will be imaged as a set of equally spaced parallel
lines in an image plane perpendicular to the optical axis, with line spacing proportional to
Equations (1) through (5) are the basic Gaussian imaging equations defining a perfect
focusing lens. Equation (2) is sometimes called the newtonian form of Eq. (1), and is the
more useful form for application to afocal lens systems.
Afocal Lenses
With afocal lenses, somewhat different coordinate system origins and nomenclature are
used, as shown in Fig. 2. The object and image space reference points RO and RE are at
conjugate image points. Since the earliest and most common use for afocal lenses is as an
aid to the eye for viewing distant objects, image space is referred to as eye space. Object
position is defined by a right-hand coordinate system (xo , yo , zo ) centered on reference
point RO. Image position in eye space is defined by coordinates (xe , ye , ze ) centered on RE.
Because afocal lenses are most commonly used for viewing distant objects, their
imaging characteristics are usually specified in terms of angular magnification M , entrance
pupil diameter Do , and total field of view. Figure 2a models an afocal lens used at infinite
conjugates. Object height off axis is defined by the principal ray angle upo , and the
corresponding image height is defined by upe . Objects viewed through the afocal lens will
appear to be magnified by a factor M , where
tan upe 5 M tan upo
FIGURE 2 Imaging geometry of afocal lenses.
If M is negative, as in Fig. 2a , the image will appear to be inverted. [Strictly speaking, since
RO and RE are separated by a distance S , the apparent magnification seen by an eye
at RE with and without the afocal lens will differ slightly from that indicated by Eq. (6) for
nearby objects.]2
The imaging geometry of an afocal lens for finite conjugates is illustrated in Fig. 2b.
Since a ray entering the afocal lens parallel to the optical axis will exit the afocal lens
parallel to the optical axis, it follows that the linear magnification m relating object height
ho and image height he must be invariant with object distance. The linear magnification m
is the inverse of the angular magnification M :
he 1
ho M
The axial separation Dze of any two images he1 and he2 is related to the separation Dzo of
the corresponding objects ho1 and ho2 by
Dze 5 m 2 Dzo 5
It follows that any convenient pair of conjugate image points can be chosen as reference
points RO and RE. Given the location of RO , the reference point separation S , and the
magnifications m 5 1 / M , the imaging geometry of a rotationally symmetric distortion-free
afocal lens can be given as
xe 5 mxo 5
ye 5 myo 5
ze 5 m 2zo 5
Equation (9) is a statement that coordinate transformation between object space and
eye space is rectilinear for afocal lenses, and is solely dependent on the afocal
magnification M and the location of two conjugate reference points RO and RE. The
equations apply (paraxially) to all object and image points independent of their distances
from the afocal lens. Any straight line of equally spaced object points will be imaged as a
straight line of equally spaced image points, even if the line does not lie in a plane normal
to the optical axis. Either RO or RE may be chosen arbitrarily, and need not lie on the axis
of symmetry of the lens system, so long as the zo - and ze -axes are set parallel to the axis of
A corollary of invariance in lateral and axial linear magnification is invariance in
angular magnification. Equation (6) thus applies to any ray traced through the afocal
system, and to tilted object and image surfaces. In the latter context, Eq. (6) can be seen as
an extension of Eq. (4) to afocal lenses.
The eye space pupil diameter De is of special importance to the design of visual
instruments and afocal attachments: De must usually be large enough to fill the pupil of the
associated instrument or eye. The object space pupil diameter Do is related to De by
De 5
5 mDo
(The more common terminology exit pupil and entrance pupil will be used later in this
Subjective Aspects of Afocal Imagery
The angular magnification M is usually thought of in terms of Eq. (6), which is often taken
to indicate that an afocal lens projects an image which is M -times as large as the object.
(See, for example, Fig. 5.88 in Hecht and Zajac.)3 Equation (9) shows that the image
height is actually 1 / M -times the object height (i.e., smaller than the object when uM u . 1).
Equation 9 also shows, however, that the image distance is reduced to 1 / M 2-times the
object distance, and it is this combination of linear height reduction and quadratic distance
reduction which produces the subjective appearance of magnification. Equation (6) can be
derived directly from Eq. (9).
ye yo / M
tan upe 5 5
5 M tan upo .
ze zo / M 2
Equation (9) is therefore a more complete model than Eq. (6) for rotationally symmetric,
distortion-free afocal lenses.
Figure 3 illustrates two subjective effects which arise when viewing objects through
afocal lenses. In Fig. 3a , for which M 5 133 , Eq. (9) predicts that image dimensions
normal to the optical axis will be reduced by 1 / 3, while image dimensions along the optical
axis will be reduced by 1 / 9. The image of the cube in Fig. 3a looks three times as tall and
wide because it is nine times closer, but it appears compressed by a factor of 3 in the axial
direction, making it look like a cardboard cutout. This subjective compression, most
apparent when using binoculars, is intrinsic to the principle producing angular magnification, and is independent of pupil spacing in the binoculars.
Figure 3a assumes the optical axis is horizontal within the observer’s reference
framework. If the axis of the afocal lens is not horizontal, the afocal lens may create the
FIGURE 3 Subjective aspects of afocal imagery.
illusion that horizontal surfaces are tilted. Figure 3b represents an M 5 173 afocal lens
whose axis is tilted 108 to a horizontal surface. Equation (6) can be used to show that the
image of this surface is tilted approximately 518 to the axis of the afocal lens, creating the
illusion that the surface is tilted 418 to the observer’s horizon. This illusion is most
noticeable when looking downward at a surface known to be horizontal, such as a body of
water, through a pair of binoculars.
Afocal Lenses and the Optical Invariant
Equations (6) and (7) can be combined to yield
he tan upe 5 ho tan upo
which is a statement of the optical invariant as applied to distortion-free afocal lenses.
Neither upo nor upe is typically larger than 358 – 408 in distortion-free afocal lenses, although
there are examples with distortion where upo 5 908. Given a limit on one angle, Eq. (11)
implies a limit on the other angle related to the ratio ho / he 5 Do / De . Put in words, the ratio
Do / De cannot be made arbitrarily large without a corresponding reduction in the maximum
allowable field of y iew. All designers of afocal lens systems must take this fundamental
principle into consideration.
A simple afocal lens can be made up of two focusing lenses, an objectiy e and an eyepiece ,
set up so that the rear focal point of the objective coincides with the front focal point of
the eyepiece. There are two general classes of simple afocal lenses, one in which both
focusing lenses are positive, and the other in which one of the two is negative. Afocal
lenses containing two positive lenses were first described by Johannes Kepler in Dioptrice ,
in 1611,4 and are called keplerian. Lenses containing a negative eyepiece are called
galilean , and will be discussed separately. Generally, afocal lenses contain at least two
powered surfaces. The simplest model for an afocal lens consists of two thin lenses.
Thin-Lens Model of a Keplerian Afocal Lens
Figure 4 shows a thin-lens model of a keplerian telescope. The focal length of its objective
is fo and the focal length of its eyepiece is fe . Its properties can be understood by tracing
two rays, ray 1 entering the objective parallel to the optical axis, and ray 2 passing through
FIGURE 4 Thin-lens model of keplerian afocal
Fo , the front focal point of the objective. Ray 1 leads directly to the linear magnification m ,
and ray 2 to the angular magnification M :
m5 2 ;
fo tan upe
M5 2 5
fe tan upo
Equation (12) makes the relationship of afocal magnification to the Scheimpflug rule of
Eq. (4) more explicit, with focal lengths fo and fe substituting for sa and s a9.
The second ray shows that placing the reference point RO at Fo will result in the
reference point RE falling on F e9, the rear focal point of the eyepiece. The reference point
separation for RO in this location is
SF 5 2fe 1 2fo 5 2(1 2 M )fe 5 2(1 2 m )fo
Equation (13) can be used as a starting point for calculating any other locations for RO
and RE , in combination with Eq. (9).
One additional generalization can be drawn from Fig. 4: the ray passing through Fo will
emerge from the objective parallel to the optical axis. It will therefore also pass through F e9
even if the spacing between objective and eyepiece is increased to focus on nearby objects.
Thus the angular magnification remains invariant, if upo is measured from Fo and upe is
measured from F e9, even when adjusting the eyepiece to focus on nearby objects makes the
lens system depart from being strictly afocal.
The simple thin-lens model of the keplerian telescope can be extended to systems
composed of two real focusing lenses if we know their focal lengths and the location of
each lens’ front and rear focal points. Equation (12) can be used to derive M , and SF can
be measured. Equation (9) can then be used to compute both finite and infinite conjugate
image geometry.
Eye Relief Manipulation
The earliest application of keplerian afocal lenses was to obtain magnified views of distant
objects. To view distant objects, the eye is placed at RE. An important design
consideration in such instruments is to move RE far enough away from the last surface of
the eyepiece for comfortable viewing. The distance from the last optical surface to the exit
pupil at RE is called the eye relief ER. One way to increase eye relief ER is to move the
entrance pupil at RO toward the objective. Most telescopes and binoculars have the system
stop at the first surface of the objective, coincident with the entrance pupil, as shown in
Fig. 5a.
FIGURE 5 Increasing eye relief ER by moving
In the thin-lens model of Fig. 5a , RO is moved a distance zo 5 fo to place it at the
objective. Thus RE must move a distance ze 5 fo / M 2 5 2fe / M , keeping in mind that M is
negative in this example. Thus for a thin-lens keplerian telescope with its stop at the
objective, the eye relief ERk is
(M 2 1)
ERk 5
It is possible to increase the eye relief further by placing the stop inside the telescope,
moving the location of RO into virtual object space. Figure 5b shows an extreme example
of this, where the virtual location of RO has been matched to the real location of RE. For
this common-pupil-position case, the eye relief ERcp is
ERcp 5
(M 2 1)
(M 1 1)
A price must be paid for locating the stop inside the afocal lens, in that the elements ahead
of the stop must be increased in diameter if the same field of view is to be covered without
The larger the magnitude of M , the smaller the gain in ER yielded by using an internal
stop. To increase the eye relief further, it is necessary to make the objective and / or the
eyepiece more complex, increasing the distance between Fo and the first surface of the
objective, and between the last surface of the eyepiece and F e9. If this is done, placing RO
at the first surface of the objective will further increase ER.
Figure 6 shows a thin-lens model of a telephoto focusing lens of focal length ft . For
convenience, a zero Petzval sum design is used, for which f1 5 f and f2 5 2f. Given the
telephoto’s focal length ft and the lens separation d , the rest of the parameters shown in
Fig. 6 can be defined in terms of the constant C 5 d / ft . The component focal length f , back
focal length bfl , and front focal length ffl , are given by
f 5 ft C 1/2;
bfl 5 ft (1 2 C 1/2);
ffl 5 ft (1 1 C 1/2)
and the total physical length ttl and focal point separation sf are given by
ttl 5 ft (1 1 C 2 C 1/2);
sf 5 ft (2 1 C )
The maximum gain in eye relief will be obtained by using telephoto designs for both
objective and eyepiece, with the negative elements of each facing each other. Two cases
are of special interest. First, ttl can be minimized by setting C 5 0.25 for both objective and
eyepiece. In this case, the eye relief ERttl is
ERttl 5 1.5
(M 2 1)
fe 5 1.5ERk
Second, sf can be maximized by setting C 5 1.0 for both objective and eyepiece. This
FIGURE 6 Zero Petzval sum telephoto lens.
FIGURE 7 Terrestrial telescope.
places the negative element at the focal plane, merging the objective and eyepiece negative
elements into a single negative field lens. The eye relief in this case, ERsf , is
ERsf 5 2.0
(M 2 1)
5 2.0ERk
Placing a field lens at the focus between objective and eyepiece can be problematical,
when viewing distant objects, since dust or scratches on the field lens will be visible. If a
reticle is required, however, it can be incorporated into the field lens. Equations (14), (18),
and (19) show that significant gains in eye relief can be made by power redistribution. In
the example of Eq. (18), the gain in ER is accompanied by a reduction in the physical
length of the optics, which is frequently beneficial.
Terrestrial Telescopes
Keplerian telescopes form an inverted image, which is considered undesirable when
viewing earthbound objects. One way to make the image erect, commonly used in
binoculars, is to incorporate erecting prisms. A second is to insert a relay stage between
objective and eyepiece, as shown in Fig. 7. The added relay is called an image erector , and
telescopes of this form are called terrestrial telescopes. (The keplerian telescope is often
referred to as an astronomical telescope , to distinguish it from terrestrial telescopes, since
astronomers do not usually object to inverted images. Astronomical has become
ambiguous in this context, since it now more commonly refers to the very large aperture
reflecting objectives found in astronomical observatories. Keplerian is the preferred
terminology.) The terrestrial telescope can be thought of as containing an objective,
eyepiece, and image erector, or as containing two afocal relay stages.
There are many variants of terrestrial telescopes made today, in the form of binoculars,
theodolites, range finders, spotting scopes, rifle scopes, and other military optical
instrumentation. All are offshoots of the keplerian telescope, containing a positive
objective and a positive eyepiece, with intermediate relay stages to perform special
functions. Patrick5 and Jacobs6 are good starting points for obtaining more information.
Field of View Limitations in Keplerian and Terrestrial Telescopes
The maximum allowable eye space angle upe and magnification M set an upper limit on
achievable fields of view, in accordance with Eq. (11). MIL-HDBK-1417 lists one eyepiece
design for which the maximum upe 5 368. If M 5 73 , using that eyepiece allows a 5.98
maximum value for upo . It is a common commercial practice to specify the total field of
view FOV as the width in feet which subtends an angle 2upo from 1000 yards away, even
when the pupil diameter is given in millimeters. FOV is thus given by
FOV 5 6000 tan upo 5
tan upe
For our 73 example, with upe 5 368 , FOV 5 620 ft at 1000 yd. For commercial 7 3 50
binoculars (M 5 73 and Do 5 50 mm), FOV 5 376 ft at 1000 yd is more typical.
Finite Conjugate Afocal Relays
If an object is placed in contact with the front surface of the keplerian telescope of Fig. 5,
its image will appear a distance ERk behind the last surface of the eyepiece, in accordance
with Eq. (14). There is a corresponding object relief distance ORk 5 M 2ERk defining the
position of an object that will be imaged at the output surface of the eyepiece, as shown in
Fig. 8. ORk and ERk define the portions of object space and eye space within which real
images can be formed of real objects with a simple keplerian afocal lens.
ORk 5 M (M 2 1)fe
Object relief is enlarged by the power redistribution technique used to extend eye relief.
Thus there is a minimum total length design corresponding to Eq. (18), for which the
object relief ORttl is
ORttl 5 1.5M (M 2 1)fe
and a maximum eye relief design corresponding to Eq. (19), for which ORsf
ORsf 5 2.0M (M 2 1)fe
is also maximized.
Figure 9 shows an example of a zero Petzval sum finite conjugate afocal relay designed
to maximize OR and ER by placing a negative field lens at the central infinite conjugate
image. Placing the stop at the field lens means that the lens is telecentric (principal rays
parallel to the optical axis) in both object and eye space. As a result, magnification,
principal ray angle of incidence on object and image surface, and cone angle are all
invariant over the entire range of OR and ER for which there is no vignetting.
Magnification and cone angle invariance means that object and image surfaces can be tilted
with respect to the optical axis without introducing keystoning or variation in image
irradiance over the field of view. Having the principal rays telecentric means that object
and image position can be adjusted for focus without altering magnification. It also means
that the lens can be defocused without altering magnification, a property very useful for
unsharp masking techniques used in the movie industry.
FIGURE 8 Finite conjugate keplerian afocal lens
showing limits on usable object space and image
FIGURE 9 Finite conjugate afocal relay configured to maximize eye relief ER and object relief
OR. Stop at common focus collimates principal rays
in both object space and eye space.
One potential disadvantage of telecentric finite conjugate afocal relays is evident from
Fig. 9: to avoid vignetting, the apertures of both objective and eyepiece must be larger
than the size of the associated object and image. While it is possible to reduce the diameter
of either the objective or the eyepiece by shifting the stop to make the design
nontelecentric, the diameter of the other lens group becomes larger. Afocal relays are thus
likely to be more expensive to manufacture than focusing lens relays, unless object and
image are small.
Finite conjugate afocal lenses have been used for alignment telescopes,8 for laser
velocimeters,9 and for automatic inspection systems for printed circuit boards.10 In the last
case, invariance of magnification, cone angle, and angle of incidence on a tilted object
surface make it possible to measure the volume of solder beads automatically with a
computerized video system. Finite conjugate afocal lenses are also used as Fourier
transform lenses.11 Brief descriptions of these applications are given in Wetherell.2
Afocal Lenses for Scanners
Many optical systems require scanners, and if the apertures of the systems are large
enough, it is preferable to place the scanner inside the system. Although scanners have
been designed for use in convergent light, they are more commonly placed in collimated
light (see the chapter on scanners in this volume, Marshall,12 and chapter 7 in Lloyd,13 for
descriptions of scanning techniques). A large aperture objective can be converted into a
high magnification keplerian afocal lens with the aid of a short focal length eyepiece
collimator, as shown in Fig. 10, providing a pupil in a collimated beam in which to insert a
scanner. For the polygonal scanner shown, given the desired scan angle and telescope
aperture diameter, Eq. (11) will define the combination of scanner facet size and number
of facets needed to achieve the desired scanning efficiency. Scanning efficiency is the time
it takes to complete one scan divided by the time between the start of two sequential scans.
It is tied to the ratio of facet length to beam diameter, the amount of vignetting allowed
within a scan, the number of facets, and the angle to be scanned.
Two limitations need to be kept in mind. First, the optical invariant will place an upper
limit on M for the given combination of Do and upo , since there will be a practical upper
limit on the achievable value of upe . Second, it may be desirable in some cases for the
keplerian afocal relay to have enough barrel distortion so that Eq. (6) becomes
upe 5 Mupo
An afocal lens obeying Eq. (24) will convert a constant rotation rate of the scan mirror
into a constant angular scan rate for the external beam. The same property in ‘‘f-theta’’
FIGURE 10 Afocal lens scanner geometry.
focusing lenses is used to convert a constant angular velocity scanner rotation rate into a
constant linear velocity rate for the recording spot of light.
The above discussion applies to scanning with a point detector. When the detector is a
linear diode array, or when a rectangular image is being projected onto moving film, the
required distortion characteristics for the optical system may be more complex.
Imaging in Binoculars
Most commercial binoculars consist of two keplerian afocal lenses with internal prismatic
image erectors. Object and image space coordinates for binoculars of this type are shown
schematically in Fig. 11. Equation (9) can be applied to Fig. 11 to analyze their imaging
properties. In most binoculars, the spacing So between objectives differs from the spacing
Se between eyepieces, and So may be either larger or smaller than Se . Each telescope has its
own set of reference points, ROL and REL for the left telescope, and ROR and RER for
the right. Object space is a single domain with a single origin O. The object point at zo ,
midway between the objective axes, will be xoL units to the right of the left objective axis,
and xoR units to the left of the right objective axis. In an ideal binocular system, the images
of the object formed by the two telescopes would merge at one point, ze units in front of
eye space origin E. This will happen if So 5 MSe , so that xeL 5 xoL / M and xeR 5 xoR / M. In
most modern binoculars, however, So Ô MSe , and separate eye space reference points EL
and ER will be formed for the left and right eye. As a result, each eye sees its own eye
space, and while they overlap, they are not coincident. This property of binoculars can
affect stereo acuity2 and eye accommodation for the user.
It is normal for the angle at which a person’s left-eye and right-eye lines of sight
converge to be linked to the distance at which the eyes focus. (In my case, this linkage was
quite strong before I began wearing glasses.) Eyes focused for a distance ze normally would
converge with an angle b , as shown in Fig. 11. When So Ô MSe , as is commonly the case,
the actual convergence angle b 9 is much smaller. A viewer for whom focus distance is
strongly linked to convergence angle may find such binoculars uncomfortable to use for
extended periods, and may be in need of frequent focus adjustment for different object
A related but more critical problem arises if the axes of the left and right telescopes are
not accurately parallel to each other. Misalignment of the axes requires the eyes to twist in
unaccustomed directions to fuse the two images, and refocusing the eyepiece is seldom
FIGURE 11 Imaging geometry of binoculars.
able to ease the burden. Jacobs6 is one of the few authors to discuss this problem. Jacobs
divides the axes misalignment into three categories: (1) misalignments requiring a
divergence D of the eye axes to fuse the images, (2) misalignments requiring a
convergence C , and (3) misalignments requiring a vertical displacement V. The tolerance
on allowable misalignment in minutes of arc is given by Jacobs as
D 5 7 .5 / (M 2 1);
C 5 22 .5 / (M 2 1);
V 5 8 .0 / (M 2 1)
Note that the tolerance on C , which corresponds to convergence to focus on nearby
objects, is substantially larger than the tolerances on D and V.
The combination of a positive objective and a negative eyepiece forms a galilean telescope.
If the objective is negative and the eyepiece positive, it is referred to as an iny erse galilean
telescope. The galilean telescope has the advantage that it forms an erect image. It is the
oldest form of visual telescope, but it has been largely replaced by terrestrial telescopes for
magnified viewing of distant objects, because of field of view limitations. In terms of
number of viewing devices manufactured, there are far more inverse galilean than galilean
telescopes. Both are used frequently as power-changing attachments to change the
effective focal length of focusing lenses.
Thin-Lens Model of a Galilean Afocal Lens
Figure 12 shows a thin-lens model of a galilean afocal lens. The properties of galilean
lenses can be derived from Eqs. (9), (12), and (13). Given that fe is negative and fo is
positive, M is positive, indicating an erect image. If RO is placed at the front focal point of
the objective, RE is a virtual pupil buried inside the lens system. In fact, galilean lenses
cannot form real images of real objects under any conditions, and at least one pupil will
always be virtual.
Field of View in Galilean Telescopes
The fact that only one pupil can be real places strong limitations on the use of galilean
telescopes as visual instruments when M  1x. Given the relationship Dzo 5 M 2 Dze ,
moving RE far enough outside the negative eyepiece to provide adequate eye relief moves
RO far enough into virtual object space to cause drastic vignetting at even small field
FIGURE 12 Thin-lens model of galilean afocal
FIGURE 13 Galilean field-of-view limitations.
angles. Placing RE a distance ER behind the negative lens moves RO to the position
shown in Fig. 13, SF 9 units behind RE , where
SF 9 5 (M 2 2 1)ER 2 (M 2 1)2fe
In effect, the objective is both field stop and limiting aperture, and vignetting defines the
maximum usable field of view. The maximum acceptable object space angle upo is taken to
be that for the principal ray which passes just inside Do , the entrance pupil at the objective.
If the F-number of the objective is FNob 5 fo / Do , then
tan upo 5
2FNob (MER 1 fe 2 Mfe )
For convenience, assume ER 5 2fe . In this case, Eq. (27) reduces to
tan upo 5
2FNob (2M 2 1)
For normal achromatic doublets, FNob $ 4.0. For M 5 3x , in this case, Eq. (28) indicates
that upo # 1.438 (FOV # 150 ft at 1000 yd). For M 5 7x , upo # 0.558 (FOV # 57.7 ft at 1000
yd). The effective field of view can be increased by making the objective faster and more
complex, as can be seen in early patents by von Rohr14 and Erfle.15 In current practice,
galilean telescopes for direct viewing are seldom made with M larger than 1.5x – 3.0x . They
are more typically used as power changers in viewing instruments, or to increase the
effective focal length of camera lenses.16
Field of View in Inverse Galilean Telescopes
For inverse galilean telescopes, where M Ô 1x , adequate eye relief can be achieved
without moving RO far inside the first surface of the objective. Inverse galilean telescopes
for which upo 5 908 are very common in the form of security viewers17 of the sort shown in
Fig. 14, which are built into doors in hotel rooms, apartments, and many houses. These
may be the most common of all optical systems more complex than eyeglasses. The
negative objective lens is designed with enough distortion to allow viewing of all or most of
the forward hemisphere, as shown by the principal ray in Fig. 14.
Inverse galilean telescopes are often used in camera view finders.18 These present
reduced scale images of the scene to be photographed, and often have built in
arrangements to project a frame of lines representing the field of view into the image.
FIGURE 14 Inverse galilean security viewer with hemispheric field of view.
FIGURE 15 Anamorphic afocal attachments.
Inverse galilean power changers are also used to increase the field of view of submarine
periscopes and other complex viewing instruments, and to reduce the effective focal length
of camera lenses.19
Anamorphic Afocal Attachments
Afocal attachments can compress or expand the scale of an image in one axis. Such devices
are called anamorphosers , or anamorphic afocal attachments. One class of anamorphoser is
the cylindrical galilean telescope, shown schematically in Fig. 15a. Cox20 and Harris21 have
patented representative examples. The keplerian form is seldom if ever used, since a
cylindrical keplerian telescope would introduce image inversion in one direction. Anamorphic compression can also be obtained using two prisms, as shown in Fig. 15b. The
adjustable magnification anamorphoser patented by Luboshez22 is a good example of
prismatic anamorphosers. Many anamorphic attachments were developed in the 1950s for
the movie industry for use in wide-field cameras and projectors. An extensive list of both
types will be found in Wetherell.2
Equation (9) can be modified to describe anamorphic afocal lenses by specifying
separate afocal magnifications Mx and My for the two axes. One important qualification is
that separate equations are needed for object and image distances for the x and y planes.
In general, anamorphic galilean attachments work best when used for distant objects,
where any difference in x -axis and y -axis focus falls within the depth of focus of the
associated camera lens. If it is necessary to use a galilean anamorphoser over a wide range
of object distances, it may be necessary to add focus adjustment capabilities within the
There are many applications where it is necessary to perform remote viewing because the
object to be viewed is in an environment hostile to the viewer, or because the object is
inaccessible to the viewer without unacceptable damage to its environment. Military
applications5 fall into the former category, and medical applications23 fall into the latter.
For these applications, instrumentation is needed to collect light from the object, transport
the light to a location more favorable for viewing, and dispense the light to the viewing
instruments or personnel. Collecting and dispensing optical images is done with focusing
lenses, typically. There are three image transportation techniques in common use today:
(1) sense the image with a camera and transport the data electronically, (2) transport the
light pattern with a coherent fiber optics bundle, and (3) transport the light pattern with a
relay lens or train of relay lenses. The first two techniques are outside the scope of this
chapter. Relay trains , however, are commonly made up of a series of unit power afocal
lenses, and are one of the most important applications of finite conjugate afocal lenses.
Unit Power Afocal Relay Trains
Several factors are important in designing relay trains. First, it is desirable to minimize the
number of relay stages in the relay train, both to maximize transmittance and to minimize
the field curvature caused by the large number of positive lenses. Second, the outside
diameter of the relay train is typically restricted (or a single relay lens could be used), so
the choice of image and pupil diameter within the relay is important. Third, economic
considerations make it desirable to use as many common elements as possible, while
minimizing the total number of elements. Fourth, it is desirable to keep internal images
well clear of optical surfaces where dust and scratches can obscure portions of the image.
Fifth, the number of relay stages must be either odd or even to insure the desired output
image orientation.
Figure 16 shows thin-lens models of the two basic afocal lens designs which can be
applied to relay train designs. Central to both designs is the use of symmetry fore and aft
of the central stop to control coma, distortion, and lateral color, and matching the image
diameter Di and stop diameter Ds to maximize the stage length to diameter ratio. In
paraxial terms, if Di 5 Ds , then the marginal ray angle u matches the principal ray angle up ,
in accordance with the optical invariant. If the relay lens is both aplanatic and distortion
free, a better model of the optical invariant is
Di sin u 5 Ds tan up
and either the field of view 2up or the numerical aperture NA 5 n sin u must be adjusted to
match pupil and image diameters. For some applications, maximizing the optical invariant
which can pass through a given tube diameter Dt in a minimum number of stages is also
If maximizing the ratio Di sin u / Dt is not critical, Fig. 16a shows how the number of
elements can be minimized by using a keplerian afocal lens with the stop at the common
focus, eliminating the need for field lenses between stages. The required tube diameter in
this example is at least twice the image diameter. If maximizing Di sin u / Dt is critical, field
lenses FL must be added to the objectives OB as shown in Fig. 16b , and the field lenses
should be located as close to the image as possible within limits set by obstructions due to
dirt and scratches on the field lens surfaces. Symmetry fore and aft of the central stop at 1
is still necessary for aberration balancing. If possible within performance constraints,
FIGURE 16 Basic unit power afocal relay designs.
FIGURE 17 Improved unit power afocal relays.
symmetry of OB and FL with respect to the planes 2a and 2b is economically desirable,
making OB and FL identical.
For medical instruments, where minimizing tube diameter is critical, variants of the
second approach are common. The rod lens design24 developed by H. H. Hopkins25 can be
considered an extreme example of either approach, making a single lens so thick that it
combines the functions of OB and FL. Figure 17a shows an example from the first of two
patents by McKinley.26,27 The central element in each symmetrical cemented triplet is a
sphere. Using rod lenses does maximize the optical invariant which can be passed through
a given tube diameter, but it does not eliminate field curvature. It also maximizes weight,
since the relay train is almost solid glass, so it is most applicable to small medical
If larger diameter relays are permissible, it is possible to correct field curvature in each
relay stage, making it possible to increase the number of stages without adding field
curvature. Baker28 has patented the lens design shown in Fig. 17b for such an application.
In this case, field lens and objective are identical, so that an entire relay train can be built
using only three different element forms. Pupil and image diameters are the same, and
pupil and image are interchangeable.
For purposes of comparison, the two designs shown in Fig. 17 have been scaled to have
the same image diameter (2.8 mm) and numerical aperture (0.10), with component focal
lengths chosen so that Ds 5 Di . Minimum tube diameter is 4.0 mm for the rod lens and
5.6 mm for the Baker relay. The image radius of curvature is about 20 mm for the rod relay
and about 2368 mm for the Baker relay (i.e., field curvature is overcorrected). Image
quality for the rod relay is 0.011 waves rms on axis and 0.116 waves rms at full field, both
values for best focus, referenced to 587 nm wavelength. For the Baker relay, the
corresponding values are 0.025 and 0.056 waves rms, respectively. The Baker design used
for this comparison was adapted from the cited patent, with modern glasses substituted for
types no longer available. No changes were made to the design other than refocusing it and
scaling it to match the first order parameters of the McKinley design. Neither design
necessarily represents the best performance which can be obtained from its design type,
and both should be evaluated in the context of a complete system design where, for
example, the field curvature of the McKinley design may be compensated for by that of the
collecting and dispensing objectives. Comparing the individual relay designs does,
however, show the price which must be paid for either maximizing the optical invariant
within a given tube diameter or minimizing field curvature.
Periscopes are relay trains designed to displace the object space reference point RO a
substantial distance away from the eye space reference point RE. This allows the observer
to look over an intervening obstacle, or to view objects in a dangerous environment while
the observer is in a safer environment. The submarine periscope is the archetypical
example. Many other examples can be found in the military5 and patent2 literature.
The simplest form of periscope is the pair of fold mirrors shown in Fig. 18a , used to
FIGURE 18 Basic reflecting periscopes.
allow the viewer to see over nearby obstacles. Figure 18b shows the next higher level of
complexity, in the form of a rear-view vehicle periscope patented29 by Rudd.30 This
consists of a pair of cylindrical mirrors in a roof arrangement. The cylinders image one axis
of object space with the principal purpose of compensating for the image inversion caused
by the roof mirror arrangement. This could be considered to be a keplerian anamorphoser,
except that it is usually a unit power magnifier, producing no anamorphic compression.
Beyond these examples, the complexity of periscopes varies widely.
The optics of complex periscopes such as the submarine periscope can be broken down
into a series of component relays. The core of a submarine periscope is a pair of fold
prisms arranged like the mirrors in Fig. 18a. The upper prism can be rotated to scan in
elevation, while the entire periscope is rotated to scan in azimuth, typically. The main
optics is composed of keplerian afocal relays of different magnification, designed to
transfer an erect image to the observer inside the submarine, usually at unit net
magnification. Galilean and inverse galilean power changers can be inserted between the
upper prism and main relay optics to change the field of view. Variants of this arrangement
will be found in other military periscopes, along with accessories such as reticles or image
intensifiers located at internal foci. Optical design procedures follow those for other
keplerian afocal lenses.
Afocal lenses can be designed with powered mirrors or combinations of mirrors and
refractors. Several such designs have been developed in recent years for use in the
photolithography of microcircuits. All-reflecting afocal lenses are classified here according
to the number of powered mirrors they contain. They will be reviewed in order of
increasing complexity, followed by a discussion of catadioptric afocal systems.
Two-powered-mirror Afocal Lenses
The simplest reflecting afocal lenses are the variants of the galilean and keplerian
telescopes shown in Fig. 19a and 19b. They may also be thought of as afocal cassegrainian
and gregorian telescopes. The galilean / cassegrainian version is often called a Mersenne
telescope. In fact, both galilean and keplerian versions were proposed by Mersenne in
1636,31 so his name should not be associated solely with the galilean variant.
Making both mirrors parabolic corrects all third order aberrations except field
curvature. This property of confocal parabolas has led to their periodic rediscovery,32,33
and to subsequent discussions of their merits and shortcomings.34,35,36 The problem with
both designs, in the forms shown in Fig. 19a and 19b , is that their eyepieces are buried so
FIGURE 19 Two-powered-mirror afocal lenses.
deeply inside the design that their usable field of view is negligible. The galilean form is
used as a laser beam expander,37 where field of view and pupil location is not a factor, and
where elimination of internal foci may be vital.
Eccentric pupil versions of the keplerian form of confocal parabolas, as shown in Fig.
19c , have proven useful as lens attachments.38 RO , RE , and the internal image are all
accessible when RO is set one focal length ahead of the primary, as shown. It is then
possible to place a field stop at the image and pupil stops at RO and RE , which very
effectively blocks stray light from entering the following optics. Being all-reflecting,
confocal parabolas can be used at any wavelength, and such attachments have seen use in
infrared designs.
Three-powered-mirror Afocal Lenses
The principle which results in third-order aberration correction for confocal parabolas also
applies when one of the parabolas is replaced by a classical cassegrainian telescope
(parabolic primary and hyperbolic secondary), as shown in Fig. 20, with two important
added advantages. First, with one negative and two positive mirrors, it is possible to reduce
the Petzval sum to zero, or to leave a small residual of field curvature to balance
higher-order astigmatism. Second, because the cassegrainian is a telephoto lens with a
remote front focal point, placing the stop at the cassegrainian primary puts the exit pupil in
a more accessible location. This design configuration has been patented by Offner,39 and is
more usefully set up as an eccentric pupil design, eliminating the central obstruction and
increasing exit pupil accessibility.
Four-powered-mirror Afocal Lenses
The confocal parabola principle can be extrapolated one step further by replacing both
parabolas with classical cassegrainian telescopes, as shown in Fig. 21a. Each cassegrainian
is corrected for field curvature independently, and the image quality of such confocal
cassegrainians can be quite good. The most useful versions are eccentric pupil. Figure 21b
shows an example from Wetherell.40 Since both objective and eyepiece are telephoto
designs, the separation between entrance pupil RO and exit pupil RE can be quite large.
An afocal relay train made up of eccentric pupil confocal cassegrainians will have very
FIGURE 20 Three-powered-mirror afocal lenses.
FIGURE 21 Four-powered-mirror afocal lenses.
FIGURE 22 Afocal retroreflector designs.
long collimated paths. If the vertex curvatures of the primary and secondary mirrors within
each cassegrainian are matched, the relay will have zero field curvature, as well. In general,
such designs work best at or near unit magnification.
Unit Power Finite Conjugate Afocal Lenses
The simplest catadioptric afocal lens is the cat’s-eye retroreflector shown in Fig. 22a , made
up of a lens with a mirror in its focal plane. Any ray entering the lens will exit parallel to
the incoming ray but traveling in the opposite direction. If made with glass of index of
refraction n 5 2.00 , a sphere with one hemisphere reflectorized (Fig. 22b ) will act as a
perfect retroreflector for collimated light entering the transparent hemisphere. Both
designs are, in effect, unit power (M 5 21.00) afocal lenses. Variations on this technique
are used for many retroreflective devices.
Unit power relays are of much interest in photolithography, particularly for microcircuit
manufacturing, which requires very high resolution, low focal ratio unit power lenses. In
the Dyson lens ,41 shown in Fig. 23a , the powered surfaces of the refractor and the reflector
are concentric, with radii R and r given by
r (n 2 1)
where n is the index of refraction of the glass. At the center point, spherical aberration and
coma are completely corrected. In the nominal design, object and image are on the surface
intersecting the center of curvature, displaced laterally to separate object from image
FIGURE 23 Concentric spheres unit power afocal
sensor (this arrangement is termed eccentric field , and is common to many multimirror lens
systems). In practice, performance of the system is limited by off-axis aberrations, and it is
desirable to depart from the nominal design to balance aberrations across the field of
The unit power all-reflecting concentric design shown in Fig. 23b is patented43 by
Offner.44 It was developed for use in manufacturing microcircuits, and is one of the most
successful finite conjugate afocal lens designs in recent years. The spheres are concentric
and the plane containing object and image surfaces passes through the common center of
curvature. It is an all-reflecting, unit power equivalent of the refracting design shown in
Fig. 9. Object and image points are eccentric field, and this is an example of the ring field
design concept, where axial symmetry ensures good correction throughout a narrow
annular region centered on the optical axis. As with the Dyson lens, having an eccentric
field means performance is limited by off-axis aberrations. Correction of the design can be
improved at the off-axis point by departing from the ideal design to balance on-axis and
off-axis aberrations.45
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