The human eye

Link: The Physics Classroom: Refraction and the Ray Model of Light  Lesson 6 - The Eye

The simplest model of the human eye is a single lens with an adjustable focal length that forms an image on the retina, or the light-sensitive bed of nerves which lines the back of the eyeball.  The eye is either relaxed (in its normal state in which rays from infinity are focused on the retina), or it is accommodating (adjusting the focal length by flexing the eye muscles to image closer objects).

The near point of a human eye, defined to be s = 25 cm, is the shortest object distance that a typical eye is able to accommodate, or to image onto the retina.

Myopia (nearsightedness)

In a nearsighted eye, the cornea is too steeply curved for the length of the eye, causing light rays from distant objects to focus in front of the retina.  Distant objects appear blurred or fuzzy because the light rays are not in focus by the time they reach the retina.  The eye is able to form images on the retina for objects that are closer than the eye's far point.

Myopia can be accommodated for through the use of a negative lens that will cause the light rays to diverge.  The power of the lens is chosen by matching the lens' focal point with the eye's far point.

Hyperopia (farsightedness)

In contrast to myopia, hyperopia occurs when the eye is too short for the power of its optical components.  In hyperopia, the cornea is not steep enough and light rays hit the retina before they come into focus.  In the case of hyperopia, light from distant objects is focused to a point behind the retina.

A positive lens can be used to correct for farsightedness.

Focal Length and Diopters

When making and selling eyeglasses, people prefer to speak of the lens power P, measured in diopters D, instead of the focal length f.  If you want to buy eyeglasses, you need to know the power of the lenses.  Focal length and power of a lens are related to each other.

D = 1/f(m)

where
D = diopters, f = lens focal length (in meters), and a "+" sign indicates a converging lens and a "-" sign indicates a diverging lens.

For two thin lenses in contact, 1/f = 1/f1 + 1/f2, and therefore power is P = Pthin(1) + Pthin(2), i.e. the powers of thin lenses in contact add algebraically.

Problem:

What is the power of a normal human eye in diopter when focusing on an object at the near point of the eye?  Assume the lens to retina distance is 2 cm.

Solution:
P = 1/f = 1/xo + 1/xi.
The object is at the near point, xo - 25 cm = 0.25 m.  The image is on the retina, xi = 2 cm = 0.02 m.
P = 1/0.25 m + 1/0.02 m = 54/m = 54 D.