*John Church*

Last month I reviewed the “lens-maker’s formula” that allows one to calculate the focal length (the inverse of the “power”) of a thin lens, provided that one knows the refractive index of the glass and the radii of the two lens surfaces. I also mentioned that the net focal length of a combination of two thin lenses close to one another or in contact is the inverse of the sum of the two powers of the separate elements.

However, if one doesn’t already have this information, there is a simple way to find the focal length of an unknown objective. This makes use of the “lens formula” (not to be confused with the “lens-maker’s formula”). I used this method as one way to determine the focal length of the H-B objective, and I described the photographic way last month.

Here’s how it’s done. I supported the objective in its cell, vertically between wooden blocks on a table in my back yard. Then I clamped a penlight about 15 feet or so away from the objective so that the light would shine through the center of the lens and approximately perpendicular to it. I then took a piece of white cardboard and adjusted it so that the image of the penlight bulb focused sharply on this screen, which I then fixed in place. I measured the distances from the penlight to the center of the objective, and from the center of the objective to the screen. The focal length was then given by the lens formula:

1/f = 1/p + 1/q

where f is the focal length, p is the distance from the light source to the objective, and q is the distance from the objective to the screen. I measured p as 170.5 inches and q as 195.5 inches. This gave 91.07 inches as the focal length, agreeing well with the other method using the physical diameter of an eclipsed moon on a direct-objective slide taken through the whole scope.

This “back-yard” method works for any pair of p’s and q’s, as long as one is careful to keep both of them farther away from the lens than what you guess the focal length might be. Otherwise you’ll get a “virtual” image, i.e. one that can’t be thrown onto a screen. In the limiting case where the “penlight” is an object at infinite distance, such as a star, 1/p vanishes and so f = q, i.e. the distance from the objective to the image. You can measure this distance directly by casting the sun’s image on a screen. This yields a good approximation of the focal length, but it’s awkward with a long refractor pointed up at the sky.

The next subject is achromatism, or color correction. Refractors have many advantages, but they are all subject to at least some degree of chromatic aberration, (i.e. the image of a bright star or planet will have at least a touch of color fringe). The usual type of astronomical objective is composed of two elements, a “crown” element of some variant of ordinary silicate glass and a “flint” element of heavier glass, typically containing lead. Flint glass disperses white light into its various colors much more than ordinary glass and is therefore used in decorative glassware such as chandeliers and crystal goblets. In 1733, Chester Moor Hall in England, not accepting Newton’s premature statement that an achromatic lens was impossible, designed the first such telescope objective by using a crown and a flint element. This allowed the objective to have the same focal lengths for two widely-separated colors of light, in the red and blue ends of the spectrum, with light in the middle of the spectrum focusing at a slightly shorter distance. Perfect achromatism across the entire visible spectrum isn’t achievable in a two-element objective, because no two kinds of glasses have exactly equal ratios of refractive indexes over this wide a range. Sometimes objectives known as “apochromats” can approach this ideal, but these usually have other disadvantages.

Before Hall’s invention, conventional single-element objectives spread light out into continuously varying focal lengths, and it was necessary to have telescopes with enormous focal lengths (“aerial” telescopes) to minimize this effect. Hall’s work was the first great advance in refracting telescopes since the early 1600’s, as it made it possible to get relatively color-free images at convenient focal ratios of f/10 or so instead of f/100 and even higher.

To design an achromatic objective with a desired total power P, we have to first know the dispersive properties of two separate kinds of glass and then solve two simultaneous linear equations to find the powers P_{1} and P_{2} of the crown and flint elements. The power of the more dispersive (flint) glass will have to be negative and that of the crown positive, but the total power P will have to be positive in order to get a real image at the focus. The equations are as follows:

P_{1} + P_{2} = P (total power = sum of the individual element powers)

P_{1}/ν_{1} + P_{2}/ν_{2} = 0 (to make blue and red light come to the same focus)

In the second equation, ν_{1} = (n_{ middle} – 1)/(n_{ blue} – n_{ red}) for the crown element and ν_{2} is the same expression for the flint element. n _{middle} denotes the refractive indexes of the respective glasses for light at a wavelength near the middle of the visible spectrum, and n_{ blue} and n_{ red} denote the values of the refractive indexes for (typically) the bright F and C lines of the hydrogen emission spectrum at 486.1 and 656.3 millimicrons, respectively. These wavelengths are often chosen because they are easily accessible in the laboratory when measuring the refractive indexes of test prisms made from the glasses that will be used to make the crown and flint elements. The Greek letter ν (“nu”) is often called the “V” number or “constringency.”

Let’s flesh out these equations with some actual numbers for the H-B objective. We already know by two separate experiments that P = 0.432 diopters (focal length = 2.31 meters or 91.1 inches). Hastings used the green iron emission line at 561.4 millimicron as the wavelength for minimum focus. I found by trigonometric ray tracing that he achieved nearly complete achromatization for the F and C lines of hydrogen for paraxial rays (i.e. those passing near the center of the objective where our simplified theory works the best). For the two glasses that he used, ν_{1} = 56.7 and ν_{2} = 36.9.

Plugging in the respective numbers to the above equations, we have

P_{1} + P_{2} = 0.432

P_{1}/56.7 + P_{2}/36.9 = 0

Resorting to our old high school math, we find required powers of 1.237 and – 0.805 for the crown and flint elements in the middle of the spectrum. In last month’s article, I noted that P_{1} had an experimentally-determined value of 1.239 diopters and P_{2} was – 0.807. The agreement is close enough to confirm that we’ve done the math correctly and that the simple achromatization equations give results adequate for the preliminary design of an objective.

A major issue still remains; however, one that resisted the efforts of mathematical geniuses such as Euler for about 30 years after the first achromat was made. Knowing the powers of the crown and flint elements isn’t nearly enough to finish designing a good achromatic objective, since for each element the first and second radii can have infinite numbers of paired values. How are we to select which pair of radii to use for each element? This choice has far-reaching consequences for spherical aberration and coma, the two most important issues after achromatization. This complex problem was finally solved in 1764 by Alexis Clairaut and his arch-rival Jean d’Alembert, two leading scientists of the French Enlightenment. I’ll talk about how they did it next month, and how closely the H-B objective satisfies their equations.

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