*By John Church*

Some years ago I analyzed the objective lens of the AAAP’s historic 6-1/4-inch Hastings-Byrne (H-B) refractor and published the results in Sky & Telescope magazine, along with a history of the telescope^{1}. The readers of Sidereal Times might be interested in knowing how the basic concepts of geometrical optics apply to this fine old instrument.

Geometrical optics starts with the idea that light consists of thin rays whose paths through transparent media can be treated by relatively simple mathematics. This approach is oversimplified in that it ignores the whole field of physical optics, where light is treated as waves rather than rays and which allows us to account for phenomena such as interference and diffraction. Diffraction causes the series of rings we see around the telescopic images of stars, with larger apertures giving smaller rings and therefore better resolution. The mathematics of physical optics is more complex than the relatively simple algebra and trigonometry we use for geometrical optics. But when it comes to lens design and optimization, geometrical optics actually works very well.

Another simplification is often used in the preliminary design of a lens, where we assume that the elements are thin in comparison with their focal lengths. This assumption is actually better than it sounds, as refractor lenses at typical f/ratios are actually thin as compared with their focal lengths. Also, the two components needed to make a color-corrected achromat are close to one another, sometimes in actual contact.

The above approach neglects color correction as well as the important field aberrations such as spherical aberration, coma, and the like. When the design of complex lenses gets farther along, we often use trigonometric ray-tracing to optimize achromatism and the actual shapes and surface curvatures of the lens elements. But for the common case of doublet refractor objectives slower than about f/10, such as the H-B lens at nearly f/15, a computer program can give excellent design results with correction of all the significant aberrations. However, this is getting away from our main story.

A converging lens is one that is thicker in the middle than at the edge. It could be either biconvex or have one surface convex and the other concave, as long as it’s thicker in the middle. Such lenses will bring light to a focus and form a real image. On the other hand, a lens that is thicker at the edge than in the middle is a diverging lens. It could be either biconcave or have one surface concave and the other convex, as long as the edge is thicker than the middle. It cannot bring light to a definite focus by itself, but in combination with a converging lens of the right focal length, it can make a lens that does form a real image. All two-element refractor lenses are like this. The H-B lens is made up of one converging and one diverging lens made of different kinds of glass, in order to provide achromatism (color correction).

The focal length of a thin lens is given by the well-known “lens-maker’s formula”:

FL = 1/[(n – 1) (1/R_{1} – 1/R_{2>})]

where n is the refractive index of the glass, R_{1} is the radius of curvature of the first surface that the light encounters, and R_{2} is the radius of curvature of the rear surface where the light emerges. With the usual convention that light goes from left to right, a radius is considered positive if its center of curvature is to the right, and negative if to the left. Thus, for a typical biconvex lens such as the front (crown) element of the H-B objective, R_{1} is positive and R_{2} is negative.

The values of these radii as determined by my measurements with a spherometer are R_{1} = + 1,336 mm and R_{2} = – 606 mm. For light near the middle of the visible spectrum, at a wavelength of 561.4 nm, the refractive index of this glass (according to measurements by Hastings) is 1.516673. Thus, applying the lens maker’s formula, the focal length of the crown element is 807 mm for light of this wavelength.

The “power” of a lens is the inverse of its focal length. Thus, a lens with a long focal length has low power, i.e. it converges light only weakly. For the H-B crown element, the power is 1/807 = 0.001239 inverse mm. In eyeglass lenses, power is given in units of inverse meters (called “diopters”) instead of inverse mm. So the H-B crown element has a power of 1.239 diopters, which is comparable to the power of weak ordinary reading glasses.

The concept of power is useful with compound lenses such as the H-B achromat, because the power of a thin combination is simply the sum of the individual powers of the elements. The final focal length will then be the inverse of the total power. (It’s important not to confuse the power of a lens with the magnifying power of a telescope with an eyepiece in place, which is the focal length of the objective divided by the focal length of the eyepiece.)

For the flint element of the H-B objective, we have the following radii and refractive index (again for the same wavelength of light): n = 1.616333, R_{1} = – 625 mm, and R_{2} = – 3,435 mm. It’s a diverging lens, because it’s thicker at the edge than in the middle. Rounding off the results to 4 figures, its focal length comes out to – 1,240 mm and its power is – 0.0008067 inverse mm, or – 0.8067 diopters. Adding up the powers of the two elements, we get a total power of 0.0004323 inverse mm (0.4323 diopters), and a focal length of 2,313 mm, or 91.1 inches.

This is fine, but what about a direct measurement of the focal length? I experimentally determined this by two methods. First, I made a direct-objective color slide during a total lunar eclipse when the moon’s diameter was known to be 30’13” and measured the image diameter at 0.801 inches with a micrometer. The focal length came out to 91.1 inches. I also set up a 30-foot long optical bench in my back yard and meas-ured the focal length by the conjugate focus method, casting the image of a penlight bulb on a screen and getting the same result. Calculating the focal length via trigonometric ray tracing, taking into account the actual glass thicknesses and separation of the elements, gives the same value to within the limits of rounding-off errors. So for a long-focus refractor with relatively thin elements, simple theory will give excellent results for the actual focal length.

One might still ask, how does a lens designer determine in advance the relative shapes of the two elements of an achromatic lens? For a given pair of glasses, many different combinations of radii could give the same total focal length for a given wavelength of light as well as acceptable color correction. I plan to discuss this in a later article, as well as how we further determine the proper element shapes in order to minimize spherical aberration and coma.

^{1} J. Church, *Sky & Telescope,* March 1979, p. 294-300.

^{2} J. Church, *Sky & Telescope,* November 1984, p. 450-451.