Some Notes on the Designing of Aspherical Magnifiers for Binocular Vision

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    Letters to the Editor

    but to the best of the writers' knowledge no at tempt has hitherto been made to design such lenses primarily for bi-nocular vision. As a result of an extended investigation into this problem, considerable progress has been made, and it is proposed to publish a full account of the results elsewhere.

    While engaged on this work, the Report on Survey of Optical Aids for Subnormal Vision1 was brought to our notice, and we were interested to see that the specification for a proposed magnifier recommended therein (p. 58) closely resembled that of one we had already undertaken, namely 6-in. focus 6-in. diameter.

    . Unfortunately the focal length of the lens recommended was not explicitly stated, and, while endeavoring to infer its value, it was found that this otherwise excellent report did not satis-factorily deal with the magnification of low power magnifiers. As stated in the report this subject is fully discussed in South-all's "M irrors, Prisms and Lenses," but a few remarks may not here be out of place.

    When designing a reading lens (for monocular or binocular vision) two measures of its magnification may be used. These we call "conventional magnification" and "effective magni-fication." In Fig. 1 an object OA of height h is situated a t distance l from a " th in" lens L of focal length f and a virtual image of OA of height h is formed at O'A' when O'L = l. An observer has his eye at E, where LE = d.

    Assuming that the unaided eye would observe h a t the least distance of distant vision D, then we can show effective magnification,

    From this formula we can deduce,

    (1) If O' is at (i.e., when h is in the focal plane of the lens)

    (2) If = 0, and l' = -D, M= (1 +D/) and is then a maxi-mum and this we call the "conventional magnification."

    In the report1 the magnification is given by M = D / but herein we have conflicting conventions, i.e., we accommodate to D without the lens, but have the eye at rest when the lens is in use. We prefer to use the second expression M = (I +D/) as this contains the reasonable convention of the eye being

    FIG, 1. Lengths to the left of the lens are measured negative, i.e., Conrady's Convention.

    accommodated to the least distance of distant vision D, both with and without the lens although clearly d = 0 is an im-possible condition in practice especially for a lens designed to be used binocularly.

    The "effective magnification" of any given lens in practice can only be determined by introducing the measured values of I' and d into the expression for M. The discrepancy re-ferred to in footnote 36 on p. 37 of the report relating to a +5.00D lens can now readily be explained. The conventional magnification is ( +D/ )=1+25 /20=21/4 as quoted by Zeiss, and not 11/4 as would follow from the use of M=D/,

    When designing magnifiers for such purposes as industrial inspection or for cases of subnormal vision such as corrected highly myopic eyes, it is desirable to arrange that the virtual image shall be in the same plane as the bench top in order that no change in accommodation takes place when glancing from objects held in the hand under the lens to objects lying on the bench not under the lens.

    In the above discussion no discrimination has been made between monocular and binocular viewing, but in the re-mainder of this communication we describe the particular defect common to most existing lenses which we set out to correct when designing primarily for binocular vision.

    A typical lens is 6-in. focus, 4y-in. aperture. An eye relief of 5 in. is chosen and an image distance of 13 in. (approxi-mately) from the eye is used as this is found in practice to suit the majority of users. This lens will thus have a conven-tional magnification of 2.7 and an effective magnification of 1.8. For various reasons a "bending" of the lens is se-lected having radii of 7y in. and 5 in. and a section of the arrangement is shown in Fig. 2.

    FIG. 2.

    This section is so chosen as to contain both eyes of the observer and the axis of the lens both because in this section aberrations are, in general, a maximum, and furthermore the use of skew rays is largely avoided. We now consider the images of the section AB of a plane object as seen by each eye separately and by both eyes taken binocularly. I t will be found that, in practice, a working distance of about three inches will automatically be assumed as by so doing the vir-tual image will be formed at approximately 13 in. from the eye, i.e., rather more than the least distance of distinct vision. The shapes and relative positions of these images may be determined either trigonometrically or by direct measure-ment and are also shown in Fig. 2. From these it will im-mediately be apparent that the two eyes are required to fuse two completely different images and that especially towards the edge of the field of view the normal relation2 between accommodation and convergence is seriously disturbed. It should also be noted that the monocular fields extend con-siderably beyond the binocular field and this leads to a sharp discontinuity of viewing conditions. Needless to say distortion is also excessive.


    V ARIOUS papers have appeared from time to time de-scribing single lens magnifiers with aspheric surfaces.

    Some Notes on the Designing of Aspherical Magnifiers for Binocular Vision

    C. E. COULMAN AND G. R. PETRIE Imperial Chemical Industries Limited, Plastics Division, Optical Development

    Department, The Hall, Welwyn, Herts, England April 25, 1949

  • L E T T E R S T O T H E E D I T O R 613

    FIG. 3.

    I t has been our object primarily to remedy all these de-fects by aspherizing one or both surfaces of the lens and the extent to which we have succeeded will, we think, be clear from a study of Fig. 3.

    This shows the effect of aspherizing the surface remote from the eyes and in this case the generator is a conic having the equation y2 = x( 15 + 10x) using Cartesian coordinates with the usual sign conventions.

    Needless to say the best results cannot be achieved by re-stricting the design to conics and details of lenses employing higher degree curves including the 6-in. focus, 6-in. aperture lens will be given elsewhere.

    1 V. J. Ellerbrook, J. Opt. Soc. Am. 36, 679 (1946). 2 H. H. Emsley and Swaine, Ophthalmic Lenses, p. 260 (Hatton Press, London, 1947). H. H. Emsley, Visual Optics (Hatton Press, London, 1949) pp, 169, 427, 429.