The Eye

Every point in front of you sends light into your eye, and the eye’s job is to bring each of those little bundles of diverging light back together at a single point on the retina. Everything from the previous chapters (vergence, refracting surfaces and lenses) comes together here.

#An optical instrument made of tissue

Light entering the eye crosses the cornea, the aqueous humour, the pupil (the opening in the iris), the crystalline lens and the vitreous before it reaches the retina. Here is a model eye, traced ray by ray:

This is Le Grand’s schematic eye, built from the average curvatures, separations and refractive indices of adult eyes, with surface shapes from Navarro’s model. Most of the bending happens at the front surface of the cornea, where the jump in refractive index is largest (Chapter 2). The crystalline lens does the rest. In all, the eye has a power of about +60 D. The cornea has about +42 D and the lens about +22 D; they don’t simply add up, because they are several millimetres apart (Chapter 5). The eye focuses distant light on a retina about 24 mm behind the cornea.

#Myopia and hyperopia

An eye whose retina lies exactly at the focus of distant light, with its lens relaxed (the lens can add power for near objects, as we’ll see), is emmetropic. Many eyes are not. If the eye is too long for its power, distant light focuses in front of the retina: myopia. If it is too short, the light would focus behind the retina: hyperopia. Clinicians describe the error by the lens that corrects it, so myopia is negative and hyperopia positive. Try it:

In this model the refractive error comes from the length of the eye, as it does in most myopia: each extra millimetre of axial length adds about 2.5 D of myopia, a figure often rounded to 3 D in the clinic. On the retina, light that is out of focus spreads into a blur patch instead of a point, and the chart beside the eye shows what that does to vision. A single dioptre of myopia is enough to drop vision from 6/6 to about 6/18 with an average pupil.

One detail about the numbers. The refractive error here is the ocular refraction: the correction needed right at the cornea, as with a contact lens. A spectacle lens sits a little in front of the eye and, as we’ll see, needs a slightly different power.

#The far point

A myopic eye can focus some light: light that arrives already diverging. Relaxed, the eye is too powerful for its length to focus parallel light, but just right for light diverging from one particular distance. That place is the eye’s far point, the farthest point it can see clearly. Light from it reaches the eye with a vergence equal to the refractive error $K$, so its distance $k$ is

For a −3.00 D eye, light must arrive with a vergence of −3.00 D, so the far point is 1/3 m, or 33 cm, in front of the eye. Move the target:

Beyond the far point everything is blurred. Nearer than it, the eye can still focus by accommodating, which we’ll come to shortly. This is why an uncorrected myope can read without glasses, often more comfortably than with them.

A hyperopic eye’s far point is behind the eye. Relaxed, the eye is too weak for its length even for parallel light, and only light already converging toward a point behind the eye would focus on its retina. No real object sends light like that, so without accommodating, a hyperopic eye sees nothing perfectly clearly, near or far.

#Correcting the eye

A spectacle lens in front of a myopic eye has one job: to take the parallel light from a distant object and make it diverge as if it came from the eye’s far point, where the eye can focus it. A lens does exactly that when its secondary focal point F′ lies on the far point. This is the far-point principle of correction. Find the right lens for each eye:

For the myopic eye a −4.00 D lens, 12 mm in front of the cornea, puts its F′ exactly on the far point, 26.2 cm in front of the cornea. For the hyperopic eye a +3.00 D lens does the same job the other way round: it sends the light converging toward its F′, 32 cm behind the eye, which is just where this eye’s far point lies.

#Vertex distance

Because the lens has to put F′ on the far point, it matters how far it sits from the eye. Slide the lens toward the cornea and its F′ slides too, so its power has to change to keep F′ in place: this is the effectivity of Chapter 5. Here are two strong prescriptions:

A spectacle lens of power $P_{\text{sp}}$ at a vertex distance $d$ corresponds to a contact lens of power

or, the other way round, $P_{\text{sp}}=P_{\text{cl}}/(1+d{P}_{\text{cl}})$, which is what the figure works out as you move the lens. A −10.00 D spectacle lens worn at 12 mm becomes a −8.93 D contact lens; a +10.00 D spectacle lens becomes +11.36 D. The difference is negligible for weak lenses but matters above about ±4 D, which is why a strong spectacle prescription should always state its vertex distance.

#Accommodation

To see something near, the eye needs more power. The crystalline lens is elastic: when the ciliary muscle contracts, the zonules that hold the lens slacken and the lens rounds up, its surfaces curving more steeply. This is accommodation. To focus a target at a distance d, an emmetropic eye needs $1/d$ dioptres of it, for example 2.50 D for a page held at 40 cm. Bring the target closer and watch the lens:

The most accommodation an eye can produce is its amplitude. At 20 it is about 12.5 D, so the near point, the closest point the eye can focus, is about 8 cm away.

#Presbyopia

With age the lens stiffens and the amplitude falls steadily, from about 15 D in childhood to almost nothing by 60. On average it is about $18.5-0.3\times \text{age}$ dioptres (Hofstetter’s formula), with a wide spread between people. Move the age slider and watch what happens to a page held at 40 cm:

Reading at 40 cm needs 2.50 D, and the average eye can still supply that until its early fifties. But no one can sustain their full amplitude for long; reading comfortably needs about half of it held in reserve, that is, an amplitude of about twice the demand. That is why near work becomes a strain from the mid-forties onward. This gradual loss of near focusing is presbyopia.

#Reading adds

The remedy is to supply the missing plus with a lens: a reading add, worn as reading glasses or as the lower part of bifocal or progressive lenses. An add moves the whole range of clear vision closer. For an emmetrope, or an eye wearing its distance correction, the range’s far end is at $1/\text{add}$ and its near end at $1/(\text{add}+\text{amplitude})$:

A 55-year-old emmetrope has about 2 D of amplitude left, so without help the eye can see clearly only beyond 50 cm. A +1.50 D add brings the range to 29–67 cm, and the book at 40 cm is in focus. Adds are usually chosen so that the reading distance uses only half of the remaining amplitude: $\text{add}=\text{demand}-\frac{1}{2}\text{amplitude}$. Too strong an add brings the range in so far that the book falls outside it.

#Pupils and pinholes

How blurred an out-of-focus eye’s vision is depends on the size of the blur patch on the retina. The patch is the cross-section of the cone of light passing through the pupil, so its angular size is roughly the pupil diameter (in metres) times the defocus (in dioptres). Change the pupil of this −2.00 D eye:

With a 1 mm pupil even this uncorrected eye reads quite well. This is the pinhole effect, and a pinhole occluder is a quick clinical test: if vision improves through a pinhole, the problem is most likely refractive. Small pupils also extend the depth of focus, one reason why presbyopes find reading easier in bright light.

#The duochrome test

Remember from Chapter 2 that the eye is slightly more powerful for blue and green light than for red. The duochrome test puts this to use when fine-tuning the sphere. The patient compares black letters on a red background with letters on a green one. The green focus lies about a quarter of a dioptre in front of the focus for yellow light, in the middle of the spectrum, and the red about a quarter behind it, so whichever is closer to the retina looks sharper:

If the letters on red look clearer, the focus still lies in front of the retina: add minus. If the letters on green look clearer, there is too much minus: add plus. When both look the same, the retina sits between the red and green foci, exactly where it should. The test works for colour-deficient patients too, because it compares sharpness rather than colour. Clinicians remember it as “RAM–GAP”: red, add minus; green, add plus.

#Recap

• The eye has about +60 D of power, two-thirds of it from the cornea, focusing distant light on a retina about 24 mm away.
• Myopia: the light focuses in front of the retina, corrected with minus. Hyperopia: behind it, corrected with plus.
• The far point, at $1/K$, is the point the relaxed eye sees clearly. A correcting lens puts its F′ on the far point.
• Vertex distance changes the power needed: $P_{\text{cl}}=P_{\text{sp}}/(1-d{P}_{\text{sp}})$.
• Accommodation adds power up to the amplitude, which falls with age (about $18.5-0.3\times \text{age}$). Presbyopia is relieved by a reading add.
• Blur grows with pupil size times defocus; a pinhole reduces refractive blur. Duochrome: red clearer, add minus; green clearer, add plus.

#Test yourself

#Further reading

• D. A. Atchison and G. Smith, Optics of the Human Eye. Butterworth-Heinemann.
• R. B. Rabbetts, Bennett and Rabbetts’ Clinical Visual Optics. Butterworth-Heinemann.
• Y. Le Grand and S. G. El Hage, Physiological Optics. Springer.
• G. Smith, Relation between spherical refractive error and visual acuity. Optometry and Vision Science 68 (1991) 591–598.