Conducted by C. L. Stong
Telescopes scarcely larger than a shoe box that are capable
of high resolution and magnification in excess of 100 diameters
have become increasingly popular in recent years. The compact
design is achieved by folding incoming rays of light with
an optical system that consists of a pair of mirrors and
a correcting lens. Few amateurs have attempted to make such
telescopes, chiefly because the optical surfaces of the
more popular designs must be ground and polished in the
form of deep paraboloids or ellipsoids - figures that tax
the skills of experienced optical workers.
This difficulty is circumvented in a telescope developed recently
by Robert J. Magee of Concord, Mass. With the mathematical
technique of ray tracing Magee designed a set of spherical surfaces
that accomplish the same objective. Although it is not necessarily
easy to grind and polish glass to a spherical figure, the surfaces
can be made by an inexperienced worker who is patient and persistent.
Magee describes the construction as follows:
"My objective in developing this design was to achieve an
optical system that would be physically short in relation to its
effective focal length. I also wanted the resolution to be limited
primarily by the wave nature of the light. As can be judged from
the accompanying diagram the physical distance between the primary
mirror and the secondary mirror is about 1.66 times the diameter
of the aperture. When the thickness of the primary mirror and
the space required for a diagonal mirror and an eyepiece are taken
into account, the overall length of the system is roughly twice
the diameter of the aperture.
"The optical system consists of a perforated primary mirror,
a two-element corrector lens (with one surface aluminized to function
as the secondary mirror), a diagonal mirror and the eyepiece.
The small two-element corrector lens replaces the full-aperture
lens of the popular Maksutov system. The combination of an achromatic
lens and a second-surface mirror that I use is known as a Mangin
"Assume that light enters the instrument from the left.
Rays reflected by the primary mirror proceed through the corrector
lens and fall on the secondary mirror. After being reflected from
this surface the rays return through the corrector lens and come
to a focus about five inches to the right of the primary mirror.
A front-surface mirror or a prism can be inserted immediately
behind the perforation of the primary mirror to divert the rays
at a right angle into the eyepiece.
"The dimensions listed in the accompanying table are scaled
for a primary mirror 4 1/4 inches in diameter. The
resolution of the system will remain diffraction-limited if all
dimensions are altered in proportion as the aperture of the primary
mirror is increased, although the aberration known as coma will
severely limit the useful field of view at an aperture of eight
"It might seem that the five optical surfaces of this small
telescope are both more costly and more difficult to grind and
polish than the single surface of a larger instrument of the Newtonian
type, which is traditionally made by amateurs. The economy of
the system results from the small size of the glass blanks and
from the modest cost of the mounting. In terms of performance
the instrument is comparable to a Newtonian telescope of the same
aperture and is far more convenient to transport and use.
"The order in which the various surfaces are ground and
polished can be varied according to the worker's preference. The
sequence I followed is not necessarily the best one. I shall describe
the operations in that order, however, so that I can point out
the pitfalls on the basis of firsthand experience.
"The scale of optical systems can in general be altered
within reasonable limits without sensibly impairing performance:
the radii of curvature, the thickness of elements and the spacing
between elements can be changed from the calculated values if
all are kept in proportion. With this requirement in mind I ground
the primary mirror first and scaled the remainder of the system
accordingly. Later I learned by experience that an error of plus
or minus 0.2 inch in the radius of curvature of the primary mirror
need not be taken into account by altering other dimensions.
"An error of this size does change the back focal length
of the optical system and the optimum distance between the primary
mirror and the corrector-lens assembly. An increase in the radius
of the primary mirror increases the back focal length. One can
compensate for such an error by adjusting the distance between
the primary mirror and the corrector lens while assembling the
elements to the mounting.
"The central hole in the primary mirror was partially cut
in the rear of the blank before the reflecting surface was ground.
The minimum diameter of the perforation is about 0.9 inch. Optically
the primary mirror is relatively fast. The focal ratio is f/2.2.
The radius of the mirror was monitored frequently during the grinding
operation by a center-of-curvature test. The test apparatus consisted
of a point source of light formed by an illuminated pinhole in
a piece of white cardboard. The primary mirror was positioned
so that the pinhole occupied a point near the center of curvature
of the mirror. When the mirror was wet, rays from the pinhole
were reflected by the mirror and converged toward the cardboard.
"The cardboard is moved toward or away from the mirror as
necessary to focus a sharp image of the pinhole on the cardboard.
The mirror is turned as required to move the image close to the
pinhole. The distance from the surface of the mirror to the image
is measured. It equals the radius of curvature of the mirror.
"The measurements are not particularly accurate during the
early stages of grinding because rough glass is not a good optical
surface even when wet. The image of the pinhole appears as a fuzzy
spot, but it becomes increasingly sharp as the grinding progresses
through successively finer grades of abrasive. A sharply focused
image can be observed at any stage of the rough grinding by slightly
polishing the glass. Make up a conventional pitch lap coated with
optical rouge or cerium oxide. Place the mirror on top of the
lap and push it back and forth about 25 times in each direction.
Small flat areas will be polished at the tips of peaks that form
the roughly ground surface. Collectively the polished areas will
reflect diverging rays from the pinhole with sufficient sharpness
for measuring the radius to within .05 inch. The finely ground
surface is polished to a spherical figure and tested with the
conventional techniques employed by amateur telescope makers.
"I next made the corrector-lens assembly, beginning with
surfaces R3 and R4. The radii of these surfaces are equal. The
surfaces are cemented together after the lenses are polished.
Lens 2, which faces the incoming light, is used as the tool for
grinding lens 1. During grinding the tool can be supported between
blocks attached to any rigid work surface. Abrasive slurry is
applied to the glass. Lens 1 is placed on top of the slurry and
ground by conventional center-over-center strokes. The excursion
of the strokes should be about a third of the diameter of the
"The radius of curvature can be monitored by the same technique
used for checking the primary mirror. If the radius becomes shorter
than desired, simply reverse the position of the blanks: turn
the pair upside down and grind lens 2 on lens 1 to increase the
radius by the desired amount. This procedure may be necessary
during the later stages of fine grinding, although experimentation
will demonstrate that some control of the radius can be exercised
by altering the length of the grinding stroke. Strokes longer
than about a third of the diameter of the blanks tend to decrease
the radius of curvature; those shorter than a third of the diameter
increase the radius. The lens is designed to be achromatic. The
image will be free of spurious color if the radius of these curves
does not depart from the specified dimension by more than .02
"I next ground and polished R5, the external surface of
lens 2. This surface is ultimately aluminized to function as the
secondary mirror of the telescope. An extra disk of glass, which
is eventually discarded, is used as the grinding tool. The radius
should be kept to within .2 inch of the specified dimension.
"Care must be taken when grinding R5 to keep the two surfaces
of the glass concentric. Rotate the glass as grinding proceeds
in order to prevent the lens from becoming wedge-shaped. Monitor
the thickness frequently by measuring the edge with a dial micrometer.
The final thickness of the polished glass should be within .025
inch of the specified dimension.
"Fortunately the thickness of the separate elements of the
corrector-lens assembly is less critical than their sum. An error
of thickness in lens 2 can be corrected by an opposite change
in the thickness of lens 1. Incidentally, the thickness of lens
1 is somewhat easier to control than that of lens 2 because the
external surface of lens 1 is flat. It is almost impossible to
avoid grinding a small wedge angle into lens 2.
"Wedge error can be described by saying that a line connecting
the centers of curvature of the two surfaces will not pass exactly
through the center of the assembled lens. The object is to keep
the error as small as possible. The effect of small errors can
be minimized when the system is assembled by masking off the periphery
of the combination so that rays of light pass through only the
symmetrically thick portion of the glass.
"The usefulness of a dial micrometer and a turntable for
checking the radius of curvature and the tendency of the glass
to become wedge-shaped can scarcely be overstated. I used a small
lathe as the turntable. A series of eight marks spaced at equal
intervals was made around the edge of the lens with waterproof
ink, The marks serve as references for clamping the lens in the
lathe consistently at the same orientation with respect to the
jaws of the chuck and also for identifying regions of the glass
that require additional grinding.
Arrangement of lenses in Robert J. Magee's optical system
Radius of curvature (inches)
R1 - 19.25
Space between primary mirror and lens 1
R2 - def
R3 - 6.88
R4 - 6.88
R5 - 12.75
Back focal length (between vertex of R1 and
Effective focal length of the system
Optical dimensions of the system
"To check the tendency of the glass to become wedge-shaped
seat the lens firmly in the jaws of the chuck and clamp it lightly.
By manipulating the transverse and cross feeds of the lathe place
the contact point of the micrometer against the glass near the
edge of the lens. Rotate the chuck slowly. The pointer of the
micrometer will remain stationary if the thickness of the lens
"The diameter of the lens and the depth of the sagitta,
or concave surface, can also be measured with the micrometer and
the turntable. When the depth of the concave surface is known
to within .0005 inch, the radius of curvature can be computed
to within .25 inch. Dial micrometers can be read easily to within
"The accompanying diagram [top of opposite page] gives the
geometry of the lenses. The sagitta is equal to the radius minus
the square root of the difference between the square of the radius
and the square of half of the diameter of the lens, or
S = R - [R2 - (L/2)2]1/2,
in which S is the sagitta, R is the radius and L is the diameter
of the lens. For example, the sagitta of a lens 1.75 inches in
diameter that has been ground to a radius of 12.75 inches is equal
to 12.75 - (162.5625 -0.7656)1/2 =0.03005 inch.
"This formula enables the worker to anticipate the depth
of the curve that will be needed to achieve a desired radius.
Conversely, I have found it helpful to measure the sagitta periodically
during the grinding operation and compute the diminishing radius
as the work proceeds. The formula is R = L2/8S + S/2.
For example, a lens 1.75 inches in diameter when ground to a sagittal
depth of .03005 inch has a radius of 3.0625/ .2404 + .03005/2
= 12.75 inches. Make the measurements with care when working with
lenses of these proportions.
"Generating the flat surface of lens 1 will doubtless require
the most patience. The project will be simplified if the worker
has access to a standard optical fiat of the same diameter as
the lens or larger. The polished surface of the lens is tested
for flatness against the standard by optical interference. To
make the test place the standard on a solid support with the fiat
side up. Rest the lens, flat side down, on top of the standard.
Separate the pair at one side by inserting a piece of tissue paper
between the glasses near the edge.
"Flood the pair from the top with monochromatic light, such
as the yellow rays emitted by the flame of a gas burner that plays
on a wick moistened with brine. Examine the reflected pattern
of light. It will consist of a grid of light and dark bands known
as interference fringes. If both optical surfaces are flat, the
fringes will form a grid of straight, parallel bands that are
alternately light and dark. Curved fringes indicate departure
"The depth of the curvature is determined by placing a straightedge
across the center of the lens in a position that joins the ends
of a complete fringe by a straight line, as the bowstring connects
the ends of a bow. Multiply by 12 the number of partial fringes
that are enclosed by the complete fringe and the straight line
to determine in millionths of an inch the approximate deviation
of the surface from flatness. The accompanying diagram depicts
the interference pattern generated by a surface that departs from
flatness by approximately 36 millionths of an inch, or three fringes.
"The external surface of lens 1 must be ground and polished
to within less than half of a fringe of flatness. A simple method
of generating a flat surface requires three glass disks of equal
diameter, one of which can be the lens. The other two disks should
be at least a quarter of an inch thick. The procedure is based
on the principle that if three surfaces consistently make full
contact when placed together in every possible combination, all
must be fiat.
"Number the edge of each disk with waterproof ink. Begin
by grinding disk 1 on 2, then 2 on 3, then 3 on 1. Next, invert
the sequence by grinding 2 on 1, 3 on 2, and 1 on 3. Return to
the first sequence and thereafter proceed alternately. Use conventional
center-over-center strokes about .3 inch long. Grind with a slurry
of No. 600 grit in water. Limit the grinding to 25 strokes per
pair of surfaces and continue until all surfaces have been fully
ground. Finish with 10 strokes per pair of surfaces.
"Prepare a polishing lap by coating one of the glass disks
with hot pitch. When the pitch cools, divide the lap into facets
about .3 inch square by cutting grooves in the pitch. Coat the
ground surface of the lens with a slurry of rouge in water, place
the coated surface on the lap and apply about a pound of pressure
for 30 minutes, or until the facets of pitch flow into full contact
with the glass. Polish the lens for about 10 minutes.
"Test the incompletely polished surface against the standard
flat without inserting tissue paper at the edge. If the grinding
has been carefully done, no more than one or two concentric fringes
will be observed. The fringes indicate that the surface of the
lens is either uniformly concave or uniformly convex.
"Exert downward pressure on the edge of the lens. If the
surface is convex, the fringes will move toward the point where
pressure is applied. If the fringes do not move, exert pressure
on the center of the lens. If the lens is concave, the radius
of the fringes will increase. To correct the curvature, cut pitch
from the edges of the grooves to reduce the area of the polishing
facets uniformly toward the center or toward the edge of the lap
as needed to flatten the surface.
"Continue polishing with the modified lap. Test the surface
frequently as polishing proceeds and alter the area of the facets
as required. If the correction is carried too far or if irregular
zones develop, make a new lap. Continue until the surface is fully
polished and flat. The operation is not as difficult as it may
seem. It requires more patience than skill. If a standard optical
flat is not available, one can be made in a matter of hours by
the procedure described in Amateur Telescope Making-Book One,
edited by Albert G. Ingalls (Scientific American, Inc., 15th printing,
"After all five surfaces have been ground, polished and
measured, finish cutting the hole through the center of the primary
mirror. Before cutting the hole, cover (and thus protect) the
polished surface with a sheet of paper coated with pressure-sensitive
adhesive. Paper so coated is available from dealers in art supplies.
The primary mirror and the concave surface of lens 2 can now be
aluminized. If the secondary mirror is aluminized, it is worthwhile
to have the flat surface of the corrector coated for low reflection,
since this surface is used twice. Usually a company that does
aluminizing will also do coating.
"The task of making the optical elements is finished by
cementing the mating surfaces of the corrector lens. This job
is simple in principle but difficult in practice. The lenses must
be thoroughly cleaned, preferably in a solution of nitric acid
or trisodium phosphate. Airborne dust must be excluded from the
"The lenses are healed to 150 degrees Fahrenheit in a water
bath, dried and cemented at this temperature. Lens 1 is supported,
flat side down, on a tabletop covered with a sheet of lens tissue.
A few drops of warm Canada balsam are poured in the center of
the concave surface. The mating surface of lens 2 is gently lowered
into contact with the cement. A downward pressure of about two
pounds is exerted on the assembly for several minutes to squeeze
out the excess cement. The excess can be cleaned from the edge
of the lenses by a cloth moistened with xylene.
"A short, snugly fitting tube is slipped over the combination
to keep the lenses centered. A pad is placed over the aluminized
surface of lens 2 and a two-pound weight is placed on the pad.
The cement will set in about two days.
"The job may not go easily. Bubbles that are difficult to
remove tend to become trapped between the lenses. It may be necessary
to alter the viscosity of the cement. For these reasons the inexperienced
worker is urged to review the portion of the article on lens making
by J. R. Haviland in Amateur Telescope Making Advanced-Book Two,
edited by Ingalls, that describes some of the tricks of using
Canada balsam cement.
"Short telescopes that employ two reflecting mirrors require
a carefully designed system of baffles to improve image contrast,
or at least to preserve it, particularly if the instrument is
to be used in daylight. The reason is not hard to find. Assume
that a bundle of rays enters the telescope at an angle such that
it just grazes the corrector lens, passes through the hole in
the primary mirror and illuminates the focal plane. The rays make
no contribution to the image because they have not fallen on the
face of the primary mirror and hence are not focused. They simply
flood the image as a veiling glare.
Geometry of spherical lens
"Such a glare cannot develop in a telescope of the Newtonian
type that includes a solid tube because the diagonal mirror that
diverts light into the eyepiece faces away from the incoming rays.
Only the rays that are reflected by the primary mirror fall on
the diagonal mirror. Instruments that have a pair of reflecting
mirrors such as those in my design can be effectively baffled
in several ways. The combination of a tubular shield that extends
forward from the center of the primary mirror and an annular diaphragm
surrounding the corrector lens is effective if the parts are properly
proportioned. The object is to make the tubular light shield sufficiently
long and the annular diaphragm sufficiently wide to prevent rays
from sources beyond the field of view from reaching the focal
Fringes indicating spherical surfase
"The baffles are installed at some cost in terms of the
brightness of the image. The shields block out the central portion
of the mirror more or less depending on their proportions. On
the other hand, the center of the mirror does not work anyway
because it is perforated and blocked by the corrector-lens assembly.
The short tubular shield that extends forward from the primary
mirror is equivalent as a baffle to extending the main tube of
the instrument about two feet.
"Incidentally, the shielding tube can be extended behind
the mirror and clamped mechanically by the end plate of the main
tube. It then serves as a peg for supporting the primary mirror.
The diagonal mirror can also be supported in the shielding tube.
A window cut in the side of the tube admits rays to the eyepiece.
I omit the details of the mechanical construction because they
will vary with the builder's taste and the contents of his scrap
"The 4 1/4-inch Pyrex blank for the primary mirror and the
lens cement can be bought from the Edmund Scientific Co. (Edscorp
Building, Barrington, N.J. 08007). A number of companies aluminize
mirrors, including the Research Service Company (1149 Massachusetts
Avenue, Arlington, Mass. 02174). Optical glass of the quality
required for the corrector lenses is available from A. Jaegers
(691A Merrick Road, Lynbrook, N.Y. 11563). Enough glass for three
sets of elements can be cut from one piece each of the glasses
listed respectively as Catalogue No. 1590 (for lens 1) and 1591
(for lens 2) in the Raw Optical Glass section of this company's
catalogue. Incidentally, when you reduce the thickness of these
glasses, care should be taken to leave ample margin for correcting
errors of cuivature and wedge angle.
"When you assemble the telescope, take pains to align the
optical axis of the primary mirror with the optical axis of the
corrector lens. Misalignment is known as decentering. A decentering
error of .005 inch can be detected but is tolerable.
"I designed the optical system by the technique of ray tracing.
This mathematical procedure can be rather tedious when it is done
with pencil and paper. Fortunately desk computers are available.
With such a computer it is possible to adapt to electronic computation
the ray-trace equations as given in textbooks on optics. It is
important, however, to find a machine that has keyboard trigonometric
functions. A desk computer is not difficult to operate, although
I urge the programmer to arrange his calculations in a systematic
manner and to incorporate a means of verifying his answers."