A compact short-focus telescope with spherical surfaces

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 mirror.

"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 inches.

"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 glass blanks.

"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 inch.

"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
Arrangement of lenses in Robert J. Magee's optical system

 

Component

Diameter
(inches)

Radius of curvature (inches)

Index of
refraction

Dispersion

Thickness
(inches)

Primary

4.25

R1 - 19.25

(mirror)

-

0.625

Space between primary mirror and lens 1

-
-
-
-

5.8

Lens 1

1.75

R2 - def
R3 - 6.88

1.517

64.5

0.489

Cement

-
-

(cement)

-

0

Lens 2

1.75

R4 - 6.88
R5 - 12.75

1.649
(mirror)

33.8

0.297

Back focal length (between vertex of R1 and focal plane)

5.032 inches

     

Effective focal length of the system

33.47 inches

     

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 is uniform.

"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 .0002 inch.

"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 from flatness.

"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, 1962).

"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 work area.

"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 plane directly.


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 box.

"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."
 

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