Lenses and focal length

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  Optical telescopes use lens to either reflect or refract light. Reflecting lens are able to focus light by using a concave surface (similar to a shallow and very smooth bowl). Refracting lens use two (or more) media which change the path of the light while the light passes through the different media. Even though these two methods accomplish the same goal (focusing light) they have different properties which require different compromises in their construction.
  The refracting method is more tolerant when the accuracy of the surface shape is considered. But this method requires a minimum of four (4) surfaces which must meet the minimum accuracy. The reflecting method requires the surface to be twice as accurate; but requires only one surface.
  Another consideration of these two methods is their physical size. The refractor requires a purity of transmission (to allow the light to pass without unwanted distortion due to impurities and voids). The reflector is more tolerant of impurities (as long as they remain beneath the optical surface). Because of these limitations, the refractor is often smaller in aperature than the reflector. This gives the advantage to the reflector to collect more light - and thus process dimmer objects.
  Refractors require multiple elements because a single element is unable to bend (refract) all colors (frequencies/wavelengths) of light equally. The reflector returns all colors equally. The refractor corrects this weakness two ways. First, it can use exotic (and more expensive) media. Second, the focal length can be increased. Thus the refractor telescope is usually longer than the reflector (of equal aperature).
  The method of creating these lens is very similar for both types. Attention to the limitations mentioned above will determine the final product. Generalities of design are formulated based on these limitations. A routine first refractor might have a focal length of 12 to 15 longer than its diameter. The first reflector will probably be 8:1 or 6:1 ratio of focal length to diameter (this ratio is called the telescopes' "f"-number). As skill is acquired, these f-numbers can be reduced for both types.
  The general flow of work is as follows: 1) rough grinding; 2) fine grinding; 3) polishing; 4) figuring; 5) testing with additional figuring to make corrections to the final product. A rudimentary description of each step will briefly follow.`
  ROUGH GRINDING - Starting with two disks of approximately identical size, a small amount of the largest abrasive (such as carborundum) is placed on top of the first disk. A small amount of water is dripped onto the abrasive. The second disk is placed over the first disk. Gravity will make the top disk become concave, while the bottom disk will become convex. Drag the top disk towards you across the bottom disk. Stop before it tips off of the bottom disk. This is a "full" stroke. Push the top disk away from you until it just reaches the same condition on the far side of the bottom disk. Do this several times. Turn the top disk a few degrees (clockwise or counter-clockwise) and step right or left. Repeat until the sound of grinding dies out. Add additional abrasive and water and continue this until the top disk shows signs of forming a depression in the center. The outer edge of the bottom disk will begin to show wear also. Try to spend equal effort in each cycle of motion (not too critical unless the time spent in a particular location exceeds a reasonably similar amount). Because of random motions and times the surface will become more uniform as time passes. Machines which repeat these motions in a regular measure can produce erroneous surfaces defects which may be difficult to correct later. The monotonous nature of this work and the loud noise generated by this process is called "hogging" the blanks. When the depression expands to the outer edge of the top disk, and the bottom disk displays wear across its entire surface you must decide if it is time to change to the next smaller abrasive. You may always return to the larger abrasive, if the work shows an uneven wear pattern. Sizes of abrasive are usually graded by counting the number of grains that fit within a given distance. Thus the larger the grain the lower the grade number of the abrasive. Carborundum can be purchased in many different sizes. The large sizes (#40, #60, etc.) roll under the disk crushing the disk's surface. Examine the surface when you decide that the fractured surface has become uniform across the entire area. Any pits or holes larger than the grains will not be removed in the next step!
  FINE GRINDING - This step uses somewhat softer abrasives (perhaps aluminum oxide), of smaller size (#480 or #600). The size will diminish down to 5 or 3 microns (very fine, and resembles talcum powder). The grains are still rolling between the two disks fracturing the surfaces. The noise is much reduced during this step - but does not completely disappear. Effort expended doing the same motions described above is much reduced. Care must be taken to reduce the "stroke" to perhaps a third of that used in the rough grinding step. Use just enough water to make a paste form. Too much water can cause the abrasive to squeeze out before it has done its job. Again, before switching to another smaller abrasive, inspect the surface or uniformity. If blemishes are found continue with that size abrasive until the areas are uniform. The surface will begin to show reflections of a lamp bulb when held nearly 50 or 60 degrees (but not perpendicular) to the lamp.
  POLISHING - A new surface much be attached to the disk which you are NOT going to use. The material attached to it is rosin or pitch. It will be heated until it is just melted (not boiled). Boiling will introduce bubbles and will make the rosin/pitch hard and glass-like. It must remain pliable and able to conform to the other disk when the two disks are placed together for an extended amount of time (ie overnight). The polishing will use an abrasive of iron oxide or cerium oxide - termed "rouge". Unlike the previous abrasives, the rouge will become imbedded in the rosin/pitch and it will "shave" or "plane" the uncoated disk. This is a vey slow process best done with a short stroke (one-quarter or less). The reflection of a lamp will begin to be seen nearly perpendicular to the surface when the disk is done. No pits or holes should be found any where on the surface. During this step the lack of water can cause the two disks to seize. Too much water, and the time to complete the polishing will become longer. Better too long than seized! The shortened stroke keeps the shape of the two disks matched and uniform. If the disks vary in shape they will not make contact everywhere across the surface. This results in a surface which is termed "dog-biscuit". If all has gone well, the finished surface will be a portion of a perfect unblemished sphere.
  FIGURING - Any deviation from a sphere is accomplished during this step. If the design can be produced using a spherical surface, then this step is unnecessary. It must be tested (next step) to prove that it is, or is not, spherical (Foucault or Ronchi test). The process to achieve another shape (conical or aconical) is complex, but always involves removing a portion of the surface to attain the desired shape. An example would be removing material from the center to attain a parabolic shape. If material is unintentionally removed from any other portion of the surface (other than the targeted area) an incorrect or distorted shape will result - requiring even more material to be removed in order to generate the intended shape. Hence frequent testing is required. An accurate and descriptive written record is very useful to help reduce the possibility of going wrong too far before you realize your errors. This step is more "art" than science. Methods to accomplish the desired shape are as numerous as the number of people doing this work. You may read about this step in a small collection of books written by people who have found tricks which helped them reach their goal. Good luck.
  TESTING - Equipment (commercial and self-constructed) is required to determine what the shape of the surface is. The original test was created by Foucault in the 19th century. It still works for many of the standard shapes. It is sensitive and accurate when correctly interrupted. Concave surfaces are particularly easy to test using the Foucault test. Convex surfaces are more challenging. Ronchi developed another test which is especially good for testing a sphere, but less useful for any other shape.
  Some basic facts resulting in a useable lens are: 1) as reflecting surfaces become more concave the focal length will become shorter; 2) likewise, as refracting surfaces become more convex the focal length will also become shorter.

How much curvature (concave or convex) is critical; so computer programs are used to simulate the combination of curves required to design the final version of the lens. The software can tell you if the design you want is attainable or even possible. As the focal length becomes shorter, the testing and figuring steps become increasingly more difficult to accomplish. Controlling these short focal lengths becomes paramount. Small f-numbers are usually avoided for use with existing eyepieces and cameras. Ratios less than 3:1 would prove woefully inadequate.