The Human Eye: Accommodation, Myopia, and Hypermetropia

Your eye is a biological camera. It has a converging lens (the crystalline lens, plus the cornea) that forms a real, inverted, diminished image on the retina — the light-sensitive layer at the back of the eyeball. The retina is a fixed distance from the lens (about 2.5 cm in an adult), so the eye can't move the "screen" like a camera focus ring does. Instead, it changes the focal length of the lens itself by physically changing the lens's shape. This process is called accommodation.

How accommodation works

The ciliary muscles surrounding the lens can contract or relax, changing the tension on the ligaments (zonules) that hold the lens in place:

• Viewing a distant object (at infinity): The ciliary muscles relax, the zonules pull the lens flat (long focal length). The lens has its weakest converging power. Parallel rays from a far object are focused exactly on the retina.

• Viewing a close object (at 25 cm): The ciliary muscles contract, the zonules slacken, and the elastic lens bulges to a more spherical shape (short focal length). The stronger converging power is needed because close objects send diverging rays that need more bending to focus on the retina.

The range over which the eye can focus is defined by two limits:

• Far point: The farthest distance at which the eye can focus. For a normal eye, this is infinity.
• Near point: The closest distance at which the eye can focus comfortably. For a young adult, this is about 25 cm (the "least distance of distinct vision"). It increases with age — at 50, the near point may be 50 cm or more, which is why reading glasses become necessary.

The eye's optics in numbers

The cornea does about 2/3 of the focusing (its refractive power is about +43 D because the air-cornea interface has a large $\Delta n$). The crystalline lens provides the remaining 1/3 and the variable focus. Total power of the relaxed eye: about +60 D, giving $f \approx 17$ mm (for an eye with axial length 24 mm, the image forms on the retina at about 17 mm behind the combined lens system). When accommodating to 25 cm, the power increases to about +64 D.

Myopia (nearsightedness)

In a myopic eye, the eyeball is too long (or the lens is too strong). Parallel rays from a distant object focus in front of the retina, not on it. By the time the light reaches the retina, the rays have crossed and diverged again, producing a blurry image.

The far point of a myopic eye is not infinity but some finite distance — maybe 2 m, maybe 50 cm, depending on the severity. Objects beyond the far point are blurry; objects closer than the far point can still be focused.

Correction: A concave (diverging) lens in front of the eye spreads the incoming rays slightly before they enter. The combined system (glasses + eye) has less total converging power, pushing the focus back onto the retina. The corrective lens has a negative power (e.g., $-2$ D, $-4$ D). The focal length of the corrective lens equals the negative of the far point distance: if the far point is 50 cm, the prescription is $P = -1/0.50 = -2.0$ D.

Hypermetropia (farsightedness)

In a hypermetropic eye, the eyeball is too short (or the lens is too weak). Rays from a nearby object focus behind the retina. The image on the retina is blurry because the rays haven't converged enough by the time they hit the retina.

The near point of a hypermetropic eye is farther than 25 cm — maybe 50 cm, 1 m, or more. Distant objects can still be focused (the ciliary muscles accommodate), but the effort is excessive and causes eye strain.

Correction: A convex (converging) lens adds extra converging power, helping the eye bring close objects into focus. The prescription is positive (e.g., $+1.5$ D, $+3.0$ D).

Presbyopia (aging-related farsightedness)

Even eyes that were perfect at 20 eventually develop difficulty focusing on close objects. The crystalline lens stiffens with age, losing its ability to bulge. The near point recedes: ~10 cm at age 10, ~25 cm at 25, ~50 cm at 45, and beyond arm's length at 60+. This is called presbyopia, and it's corrected with reading glasses (converging lenses).

Bifocals combine a diverging upper zone (for distance, if the person is also myopic) with a converging lower zone (for reading).

Worked example: prescribing myopia correction

A student's far point is 40 cm. What lens prescription is needed?

The corrective lens must take parallel rays ($d_o = \infty$) and make them appear to come from 40 cm (the eye's far point). So the lens must form a virtual image at 40 cm from the eye.

$\dfrac{1}{f} = \dfrac{1}{d_o} + \dfrac{1}{d_i} = \dfrac{1}{\infty} + \dfrac{1}{-0.40} = -2.5 \text{ D}$

Prescription: $-2.5$ D. A lens with focal length $f = -40$ cm.

Worked example: prescribing hypermetropia correction

A person's near point is 1 m. They want to read at 25 cm. What lens is needed?

The lens must take light from 25 cm and make it appear to come from 1 m (the eye's near point).

$\dfrac{1}{f} = \dfrac{1}{0.25} + \dfrac{1}{-1.0} = 4 - 1 = 3 \text{ D}$

Prescription: $+3.0$ D. A lens with focal length $f = 33.3$ cm.

Common mistakes

• Thinking myopia is caused by the lens being too strong. It can be — but more commonly, the eyeball is too long. The lens is fine; the geometry is wrong. Same correction either way.
• Confusing near point with near-sightedness. A myopic person has a closer near point than normal (they can focus on very close objects without strain). Their far point is the problem — it's too close. "Near-sighted" means "can see near, can't see far."
• Forgetting that the corrective lens position matters. Technically, the formulas assume the corrective lens is at the eye. For glasses (which sit ~1.5 cm in front), there's a small correction. For contact lenses (which sit on the cornea), the formula is exact.

Try it in the visualization

Select "Normal eye" to see accommodation in action as you move the object from infinity to 25 cm. Switch to "Myopic" and move a distant object — watch it focus in front of the retina. Toggle "Add corrective lens" and see the diverging lens push the focus back. Do the same for "Hypermetropic." Adjust the prescription power slider to find the exact correction that puts the image right on the retina.