Eye Lens Accommodation - 2D Ray Diagram

Setting up the lab bench...

Presets

Live readouts

Graphs

Accommodation required against object vergence (red = now) and the amplitude left at this age

Retinal blur (visual angle) against object vergence

Accommodation amplitude against age (Hofstetter mean)

Spectacle power that corrects the far view against axial length

What is happening now

Starting.

Numerical checks

running...

How to use

Move the object vergence slider (1/d: 2.5 D is a book at 40 cm) and watch the lens thicken while the ray fan stays focused on the red retina. Then: (1) raise the age to 50 or more until the lens can no longer thicken enough and the letters blur (presbyopia); (2) lengthen the eyeball past 23 mm and look far away (myopia), then press Prescribe the right glasses; (3) shorten it below 22 mm (hyperopia): the young eye hides the error by accommodating; (4) untick the automatic accommodation to see the relaxed eye; (5) add reading glasses to a 55 year old eye. The green bar at the bottom shows the distances this eye sees sharply.

The science

Vergence. Light from an object at distance d reaches the eye with vergence −1/d (dioptres, metres). The eye adds its power P and the light converges to an image at distance v inside the vitreous (refractive index n = 1.336): n/v = P − 1/d. The image is sharp when v equals the axial length L, so the power needed is P_req = n/L + 1/d. A relaxed normal eye has P = 60 D, so L = n/P = 22.3 mm.

Accommodation. The ciliary muscle contracts, the zonule fibres slacken and the elastic lens rounds up and thickens, adding A dioptres: P = 60 + S + A, with S the spectacle power. The eye uses A = P_req − 60 − S limited to 0 … A_max, with the Hofstetter mean A_max = 18.5 − 0.30 · age. The near point is about 1/(A_max + 0.25 D).

P_req = n/L + 1/d A = clamp(P_req − 60 − S, 0, A_max) b = D_p L |ΔP| / n

Blur. If the power is wrong by ΔP the rays from the two pupil edges meet in front of or behind the retina, and a point becomes a disc of diameter b = D_p L |ΔP| / n. The eye accepts ±0.25 D (depth of focus) before things look blurred. The red bar on the retina and the letters in the inset use this blur.

Eye length and glasses. A long eye (myopia) focuses in front of the retina and needs a concave lens S = n/L − 60, about −2.7 D per extra millimetre. A short eye (hyperopia) focuses behind the retina and is hidden by accommodation; presbyopia is the loss of A_max, treated with a plus lens.

Honest simplifications. A reduced eye: all refraction at one principal plane in one medium, thin spectacle lens at the cornea (the drawn spectacle position is pictorial), no spherical aberration, astigmatism, chromatic effects or Stiles-Crawford effect. The globe and lens are drawn schematically (lens thickness 3.6 mm + 0.045 mm per dioptre); the object is placed on a compressed logarithmic scale. Hofstetter amplitude and the 0.25 D depth of focus are typical values, individuals vary by about 2 D.