Compound scope lenses: Strydom 2010 study

This is a theoretical paper, written by optometrists, on how magnifying lenses work in compound archery. There are no measurements and no participants: the authors apply the formulas of optics to two set-ups. In the first, with one lens, there is only the lens in the sight's scope, and the second optical element is the eye itself with its uncorrected refractive error. In the second, with two lenses, another lens sits in the plane of the peep and the pair works as a small telescope.

With one lens the result depends almost entirely on the archer's eyesight. For an eye with no refractive error the formula gives no magnification at all; for a 1 dioptre myope, with the lens at 0.80 m, a -5 D lens would be needed, which makes the target smaller (magnification 0.20); for a 1 dioptre hyperope the authors give a +0.50 D lens and a magnification of 2, with a sharp image only if the eye can accommodate. With two lenses draw length matters: with a +2 D front lens, an archer with a 29-inch draw (lenses about 0.80 m apart) would need a 3.33 D lens in the peep and would get a magnification of -1.67, meaning an inverted image; with 24 inches (about 0.4 m) a -10 D lens would be needed, giving a magnification of 5. The conclusion is that the choice has to be made for the individual, taking account of age, refractive error, draw length, pupil and peep diameter and how much magnification is needed, and bearing in mind that more magnification makes aiming less steady.

The authors use two pairs of formulas. For the one-lens system, the distance between lens and eye is d = 1/F1 + 1/Fs and the magnification is M = -Fs/F1, where F1 is the power of the lens in the sight and Fs the refractive error of the eye, in dioptres. For the two-lens system the same relations link the sight lens (the objective, F1) to the peep lens (the eyepiece, F2), with d equal to the distance between peep and sight, which depends on draw length. A negative magnification means an inverted image, one below 1 a reduced image.

The rest of the paper discusses, in qualitative terms, the factors that affect sharpness: pupil and peep diameter (the pupil gives the best acuity between 2 and 5 mm), depth of field and depth of focus, accommodation and presbyopia, which usually appears around the age of 40, and the option of moving the lens further out with the sight's extension bar. The work comes from a master's thesis in optometry at the University of Johannesburg.

There are no measurements at all: not on archers, not on real lenses, not on scores. The figures in the examples come from the formulas, and some of them do not quite add up. Using the same formulas, for the 1 dioptre hyperope with the lens at 0.80 m one gets about +0.56 D and a magnification of 1.8: the values reported (+0.50 D and 2) correspond to a distance of 1 metre. The 0.4 m distance given for a 24-inch draw, five inches shorter than the archer at 0.80 m, is not explained. For the eye with no refractive error the formula gives zero magnification, a result the authors themselves then qualify in words by bringing in accommodation.

The model uses thin lenses and takes no account of the quality of real lenses or of aberrations. The idea that magnification makes aiming less steady comes from an earlier study and is reported without figures. Some statements rest on personal communications, and the paper does not go into what competition rules allow for a lens in the peep.

For a compound archer the paper makes two things clear. The first is that with a single lens in the scope the effect depends on eyesight: the same lens can magnify for one archer, shrink the target for another and blur it for a third, so the power that works for a fellow archer on the line says little if their eyesight is different. The second is that, for a given lens, the distance between eye and sight matters: draw length and the extension bar change the magnification, and a short draw with a strong lens can give more magnification than an archer can handle while aiming.

For archers with a refractive error, the authors suggest starting from the best distance correction, with glasses or contact lenses, and choosing the sight lenses on that basis; after 40, with presbyopia, the choice between a sharp target and a sharp pin becomes more delicate, and a smaller peep increases depth of field, as long as the pupil does not fall below 2 mm, where diffraction reduces acuity. These are principles drawn from optics, not results measured on archers, and they assume an eye examination by an optometrist.