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Foreword

The National Bureau of Standards receives many requests for information
on the necessary procedures for designing telescope objectives as well as for
copies of designs themselves. There are very few such designs published where
they are accessible to an amateur, their characteristics are rarely adequately
described, and they may require glasses that are no longer obtainable. There
are many amateur telescope makers who would like to compute their own de-
signs for refracting-telescope objectives. Although there are several books in
existence that describe procedures of design, most of them require the reader
to review considerable theory before they get to the how-to-do-it part.

It is the purpose of this Circular to present a procedure and the necessary
equations in such a way that anyone familiar with elementary algebra can apply
this procedure and in a relatively short time develop the proper curves for a
well corrected telescope objective. The method is also applicable to the
design of low-power microscope objectives.

A. V. ASTIN, Director.

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1025-54

Computation of Achromatic Objectives

Robert E. Stephens

Procedures for the computation of the curves for achromatic doublet objectives by the algebraic method are presented in considerable detail, using the design of two such objective as examples. The reasons for choosing particular pairs of glass types and the changes in procedure necessary to accomplish different degrees of correction are discussed.

The information contained in this Circular is sufficient to guide the reader in computing algebraically achromatic objectives for his own specific applications. It also serves as a introduction to the algebraic method for those who may wish to delve further into the design of lens systems.

1. General Considerations

Telescope objectives usually consist of only two elementary lenses, one of crown glass and one of flint glass. The crown glass practically always faces the incident light, although equally good objectives can be made with the glasses in the reverse order. Although three or more elements are occasionally used in special purpose objectives to achieve a large relative aperture, this paper is restricted to the design of doublets, and the equations given are simplifications of the more general ones and are applicable to this special case of doublets only.

Perhaps the shortest procedure for computing telescope objectives in which the aberrations are either corrected or reduced to small prescribed values is the combination of the elegant but approximate thin-lens algebraic aberration equations with the relatively more exact, but more tedious, procedure of ray tracing. A satisfactory set of glasses and a preliminary design may be arrived at much more quickly by the algebraic method than by ray tracing. This preliminary design is then tested by ray tracing and modified slightly to achieve optimum corrections.

It is convenient to compute a design for unit focal length and scale the resulting design to any desired focal length. It is also convenient to extend the thin-lens approximation to the combination of two lenses, assuming a complete system of zero thickness. As suggested above, the designs obtained on such assumptions are not usually final but may be modified slightly by addition of thicknesses and sometimes air spaces, and by small changes in the curves. The modifications are made as a result of analysis by means of exact ray tracing.

With two elements at the disposal of the designer, an objective may be computed that has the chromatic aberration, the spherical aberration, and the coma either corrected or reduced to suitable small residuals. To accomplish such correction it is

296264-54

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where is the lens power, f the focal index of refraction of the glass, and the first and second radii of curvature, The signs of the quantities used are with the usual conventions of coor etry, which is different from the con in some elementary physics texts b simpler to use. The light is thought ing from left to right, the positive radius is measured along the axis of line connecting the centers of curvatu center of the lens surface to the center It is positive if the center of curvat right of the surface, negative if to th double-convex lens the first radius is second is negative.

The above equation may also be (n-1)K, where K is called the total c is defined by K= (1/ra) — (1/ro). If any arbitrary value be chosen fo then a corresponding value for r, that desired value of K. Thus the shape be changed without changing the po ing the shape of a lens changes the n the spherical aberration and coma. by choosing the correct shape to red of a simple lens to zero, but the sph tion can, in general, only be reduced t

in this way. The following quanti

shape factor, is used in the equations aberration and coma: σ= = (ro+ra)/(ro|

To illustrate what this means, two s whose total curvatures are the san have different shapes are shown in fi

The following equations, which ar of shape factors, are used to determi

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First, two suitable glass types are chosen. This is partly by experience, but almost any pair of glasses whose Abbe numbers differ by 20 or more will make a satisfactory objective. For this example, borosilicate crown, n=1.519, V=64.5, is used for the first element, and dense flint, ne= 1.6252, V=36.2, is used for the second element. The subscript e refers to the wavelength 5461 angstroms (mercury green). Glasses with these characteristics are made by practically all American makers of optical glass.

Next the required lens powers are computed by means of eq (1) and (2). In order to obtain full correction of the chromatic aberration we let chr=0. The following powers are obtained:

41=+2.279

2=-1.279.

We next compute the positions of image and object for each element by means of the equations

and

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(3)

or

(4)

ere the A's and C's are functions of the refrace indexes, the shape factors, and the positions image and object. They will be defined later en the need for them arises.

. Procedure for Calculating a Design

The procedure for calculating a design will be tlined in the following by the actual calculation an example. This example is an uncemented romat, with full correction of spherical aberion and coma. This type of lens is adapted to e in high-power astronomical telescopes of 3-in. greater apertures, where a cemented objective impracticable because of the difference in the ermal expansivities of the two types of glass.

d

For this example,

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