ACHROMATISM AND THE MAKING OF REFRACTING TELESCOPES.
The index of refraction of glass is a measure of the amount that light is refracted or bent in passing through a lens or prism made from this glass. It is not the same for light of all colors; red being refracted less than blue and each color somewhat differently. So that light rays, after passing through a lens, will not all come to the same focus, the focus of the blue being nearer the lens than the focus of the red. Since ordinary light is a combination of many colors, you can readily see that a single lens would not make a good telescope objective.
The difference between the refraction of red and blue light by a certain kind of glass is known as its dispersion. This is not the same for all kinds of glass; a flint-glass lens or prism dispersing the light about twice as much as similar lens or prism of crown glass. An achromatic lens is made by combining a positive lens of crown glass with a negative lens of flint glass which has the same amount of dispersion as the crown. Since the flint glass is negative, its dispersion is opposite to that of the positive crown lens; and when we combine them their dispersions are neutralized and the red and blue rays come to the same focus. (Fig. 2) The combination will be positive; for, owing to the high dispersive power of the flint glass, the negative lens will be only about half as powerful as the positive lens when their dispersions are equal in amount.
In calculating an achromatic lens a great deal of advanced mathematics may be used; but we shall do very well by certain simple rule of thumb methods.
First we must get our crown and flint glass. It is well to use the kinds of glass recommended by the dealer as suitable for telescope objectives. With the glass he will supply that index or coefficient of refraction for each kind of glass, and another characteristic known as the V value. If the dispersion is not given separately we find it by subtracting 1 from the coefficient of refraction (ND) and dividing the remainder by the V value - Dispersion = (ND - 1)/V.
For the sake of simplicity we will make our objectives with both curves of the crown glass, and the first curve of the flint glass, equal - R1 = R2 = R3. If we make the radius of this curve 0.4 of the desired focus of the achromatic lens, we will get very nearly the desired focus for the combination, when we have given our last surface a curve that will produce an achromatic combination.
If the dispersion of the flint is just twice that of the crown glass, the last surface of the flint lens will be flat; if less, it will be concave; and if more, convex.
The radius of curvature for the fourth surface is found by multiplying the radius of the first surface by the dispersion of the flint glass, and dividing the product by the difference between the dispersion of the flint and twice that of the crown. In Fig. 1 the computation for a lens of 12" focal length is carried out.
After the lenses have been ground, polished, and centered, as described in Hobbygraf #1, they are cemented together with Canada Balsam, dissolved in xylol or acetone, which makes a permanent joint. A little of the balsam is placed between the lenses, which are then placed on a sheet of asbestos and heated over a gas or alcohol flame until the balsam flows out all around the edges. They are then allowed to cool. The joint should be perfectly transparent. If there are imperfect places, reheat, separate them; clean with turpentine and then soap and water; and try again.
Fig. 3 shows a finder telescope with an achromatic objective. As in all telescopes, the focal plane of the objective should coincide with that of the eyepiece. The cross wires are also placed at this common focus. To make the cross wires, cut out a washer-shaped diaphragm, large enough so that when pressed into the telescope tube it will remain where placed. Punch or drill four small holes equally spaced, and thread through them a fine copper wire crossing at the center of the washer. Blacken the wires by heating them with a match. A single-lens eyepiece is shown, but of course any kind may be used. If it is a Huyghenian, the cross-wires should be placed on the diaphragm between the two lenses of the eyepiece. This may be pushed from the eyepiece tube and replaced after fitting with wires. If the eyepiece is of the Ramsden type, the cross-hairs are placed in front of the field lens of the eyepiece. In any case the cross-wires should be in perfect focus when observed through the eyepiece.
The foregoing instructions are applicable to small objectives only. Where the diameter is over 2", it is advisable not to cement the crown and flint lenses together, but to separate them slightly. The amount of separation is determined by trial; that separation giving the least color to the image being selected. A correction for Spherical Aberration is also made, where the best definition is necessary.
The making of a 4" objective will now be described. The methods used are applicable to larger or smaller sizes.
A focal length of about 15 times the diameter of the lens has been found to give the best results. It avoids the excessive spherical aberration of shorter focus lenses, and is not so long that the illumination of the planets or nebulae suffers unduly.
We decide on a 15 x 4 = 60" for the focal length of our objective and compute the curves as described for the smaller objective in the first part of this article.
We will run through these computations for an objective made from Jena [Historical Note: Jena was a major supplier of glass before WW2 but ended up behind the Iron Curtain after the war as it was in the East part of Germany which the Russians controlled. Bob May] glass with the following characteristics:
Nd = 1.516 V = 64.0 d = 0.0081
Nd = 1.620 V = 36.4 d = 0.0171
R1 = R2 = R3 = .4 X 60 = 24".
R4 = (R1 x d1)/(d1 - 2d)
= (24 x 0.0171)/(0.171 - 0.0162)
= (24 x 0.171)/0.0009 = 456".
Since the dispersion of the flint is more than twice that of the crown this last surface is convex. Surfaces 1, 2, 3 are all of 24" radius, 1 and 2 being convex and 3 concave. Surface 4 has a radius of 456" and is convex.
A lens of this size or one requiring extreme accuracy should never be ground or polished on a rotating lap. It has to be hand-ground and polished in the same manner as the concave surface of the mirror of a reflecting telescope (see Hobbygraf #11).
Besides the crown and flint disks we must have two plate glass disks of the same size for tools. Place the crown disk on the workbench and fasten it by driving three nails around it. Hole one of the plate glass disks in the hand, and, using the stroke and motions used in making a concave mirror, rough grind the crown disk to a curve of 24" radius, as determined by applying a template to the curve.
The template is made as follows: Drive two nails through a wooden strip 24" apart. One nail is driven through into the workbench, and serves as a pivot on which to turn the strip. A piece of tin or sheet zinc is placed under the other nail, and a curve of 24" radius is marked on the sheet metal, which is then cut out to serve as a template.
When surface 1 is rough-ground to fit the template, the crown disk is turned other side up and ground with the flint disk as a tool to the same curve. It is advisable to lay the crown in a wooden circle, or to gouge out the board under it so that it will lie firmly and not rock on the surface which is first ground.
As rough-grinding nears completion, remove the crown disk from the bench and measure the edge with a caliper made by filing a notch in a piece of sheet brass to fit the edge of the lens as shown in Fig. 5. You will probably find one part of the edge thicker than the other; and this must be remedied by turning the edge of the lens toward you and pressing harder at the beginning of the stroke in grinding. When the edge measures the same all around, you are ready to fine-grind.
Fine-grind as in making a concave mirror, (Hobbygraf#11) using a short stroke (about an inch from end to end) with graduated grades of abrasive from the coarsest #2 to the finest #6; and polish on a pitch lap with optical rouge in the same manner as a mirror. The center of the lap should be reduced in area to encourage the oblate spheroidal figure much easier (Fig. 6). Also, be sure to trim the edge of the lap, making it about 3 3/4" in diameter for the 4" lens, to avoid a turned down edge.
It is immaterial whether the lens or the lap be held in the hand; but, if the lens is on the table, it should be turned at frequent intervals; or, better, hold it on a barrel or pedestal so that the worker can move completely around it while find grinding and polishing.
Before starting to polish, check with the edge gauge and be sure that the edge is the same thickness all around. Place clean paper under the lens, to avoid scratches on the finished surface. After polishing the crown lens, a new lap - convex - is made and the third surface polished.
Next, a new tool is selected, and the fourth surface ground and polished. We found that it is to be convex, with a radius of 456". Using our old formula: depth = r^2 / 2R we get 4/932 or 0.0044" for the height of the convexity (or the depth of the hollow of the tool). This is just about the thickness of the cover of a magazine (circla 1933); so, placing the flint disk on the workbench with the fourth surface up, grind until you can just slip a corner of a magazine cover under a straight edge laid across the tool.
For this slight depression it is better to rough-grind with #2 grit - 120 grain - rather than the coarser #80 grit ordinarily used. If the disk is badly wedge-shaped, as they sometimes are, you can save time by grinding it down to an even thickness before starting. In any case, the flint should be of the same thickness as the crown before starting. When fine-grinding it completed the edge should be the same thickness all around as tested with an edge gauge or micrometer. Then polish the fourth surface. This completes the lens, except for adjustment and figuring the fourth surface for spherical aberration.
Before carrying out the final corrections, we will make a cell and tube for our objective. Make the cell and ring from aluminum or brass on a lathe, as shown in Fig. 7. It is made extra deep, so that the two lenses may be separated if it should prove desirable, to improve the achromatism of the objective.
The tube may be made of brass 5" in diameter. The counter cell is fastened permanently to the tube. The cell is removable, and had three pairs of push-pull screws to square it on the axis of the tube.
Place the lenses in the cell with a ring of heavy paper between them. Do not drill the retaining ring for screws until later. The friction between the ring and the cell should be sufficient to hold the lenses in place while testing. YOu may also leave the push screws out of the cell flange until ready to square on the objective.
Assemble the tube and counter cell and the cell with its contained objective; and prop it up on a table so that it points at some distant light - at least a city block distant. The smaller the light and the greater the distance, the better. A star would be best if it would keep still long enough. Looking at this light through the tube, move the head back until the light expands and covers the entire objective and at this place setup your knife-edge for testing by the Foucault method; just as you did with your mirror, as explained in Hobbygraf #11. [Note: The edge of the light will act as the slit as it does in the slitless tester design with a mirror. Bob May]
Adjust to give the characteristic shadows of the Foucault test. If the objective darkens evenly all over when the knife-edge cuts the cone of the rays, your objective is perfect, so far as spherical aberration goes at least; but probably you get the parabolic or oblate spheroidal shadows. These must be eliminated by polishing the fourth surface of your objective to give the flat shadow that, in the case of the mirror, indicated a spherical curve.
If the shadows indicate a hill at the center, remove the flint lens and polish the 4th surface with a long stroke until the correction is mad as indicated by replacing the lens and testing as before.
If, in stead of a hill, you have a hollow, remove a star-shaped area from the center of the lap and polish until the hollow disappears. If there is doubt as to whether it is a hill or a hollow, touch the lens at the center with the warm hand for a minute. This will cause that area to expand with the heat from the hand. Then observe the shadows and, if the effect is increased, it was a hill. If, on the contrary, the expansion has counteracted the defect observed (so that the defect is reduced or has entirely disappeared) you know it is a hollow and can act accordingly. If there was a turned-down edge, reduce the diameter of the lap and polish until this effect disappears. You may also have to repair the other 3 surface to get rid of this defect.
If the effects are too great, it may be necessary to correct other surfaces besides the fourth. If, after several hour's work, you have made little progress, try another and another until you have produced an objective free from spherical aberration - one that darkens evenly all over as the knife-edge cuts in at the focus.
If you are so fortunate as to have an optical flat as large as your lens, you can use the Auto-Collimation method of testing, which allows the entire process to be done indoors. See Fig. 4. This shows the setup for testing a lens, using the pinhole star and the knife-edge. The flat, previously silvered, is setup close to the objective, and the lamp and knife-edge are placed on either side of the focus, in this case about 60" from the objective. Testing and correction are carried out exactly as in the case of a distant light. Light from the pinhole, near the focus, passes through the objective lens and leaves it as parallel rays which are returned by the silvered flat to the objective, which now brings them to a focus on the knife-edge.
Before the tests and adjustment for achromatism the telescope should be completed by constructing and putting in the eyepiece slide, as shown in Fig. 8. The distance from the objective to the focal plane of the instrument is the distance from the objective to the knife-edge in testing on a distant light. Measure this distance and make the reducer tube of sufficient length to extend to within 1/2" of the focal plane. The eyepiece slide carries the eyepiece and is used to focus it.
One form of mount must be provided before proceeding further; that shown in Hobbygraf #9 is very good. The pedestal should be about 6 feet high, for comfort in observing objects near the zenith.
We are now ready to square on the objective; that is to adjust it so that its optical axis passes through the center of the eyepiece. To do this we point the telescope at the North Star and focus very carefully. The North Star is chosen because it will not move rapidly out of the field of sight, as stars further south do, unless the instrument is moved to follow them.
You will probably find that, instead of a small round point of light, the star image is fan or pear shaped; the smaller end being the brighter. This indicates that the objective is too near the eyepiece on the side corresponding to the small bright part of the image. With a piece of crayon mark the telescope tube on this side; then, using the push-pull screws, move this side of the cell further away from the counter cell (or the opposite side nearer) and look at the star again. Repeat until the image shows as a round dot. If the image is thrown a little out of focus by moving the eyepiece in or out, it should expand into a perfectly circular are mad up of concentric rings.
Sometimes several of the lens surfaces may be astigmatic; that is, the radius of curvature may be different along different diameters of the lens. This condition is shown by elliptical images and rings when the image is thrown out of focus. The remain elliptical even when squaring on is as perfect as can be accomplished. To correct this appearance, try revolving the flint lens on the crown, testing it on a star in a number of different relative positions, and choosing that position which gives the most truly circular image and the ring system. When this is found, make a pencil mark across the edges of both lenses so that they can always be replaced in this position.
When the objective is thus adjusted for astigmatism, and perfectly squared on, we can examine the image for achromatism.
In this, as in squaring on we use a 1/4" EFL eyepiece and examine the image of Polaris. If the objective is a good one the image should be greenish yellow in color surrounded by a very faint violet halo.
Different separations of the crown and flint may be tried, by changing the thickness of the spacer between the lenses; and that separation is chosen permanently which give the image with the least halo around the star. Be sure to mark the cell and counter cell, and also the edges of the crown and flint lenses, so that they may all go together in the same way each time you assemble them.
Now drill and tape for the screws that hold the lens retaining ring. Screw it in place, reassemble - and your telescope is complete. You will, some time, want a rack and pinion focusing device instead of the simple slid described, and will probably elaborate and perfect your mount as time goes on. However, the simple instrument described will give excellent service, and you should not attempt too pretentious an outfit to start with. Let your telescope grow as your knowledge expands and as you learn what you really want and need.