BACKGROUND AND THEORY

Until the 17th century it was generally thought that light was a stream of particles (known as corpuscles). Huygens proposed that many phenomena could be explained much more satisfactorily by assuming the light was in fact a wave. Maxwell (1873) demonstrated that light is a part of the electromagnetic spectrum which in our common experience covers:

X-Rays Wavelengths approx 10^-10 m

Ultra-Violet 10^-8 m

Visible Light 5 x 10^-7 m

Infrared 10^-5 m

AM Radio Broadcast Band 300 m

All forms of electromagnetic radiation travel at the same speed (the speed of light) which is 3 x 10⁸ m/sec.

When a parallel beam of light is incident on a plane surface of a material which is transparent (e.g. glass or water) two beams are formed as shown in fig. 1. The reflected and refracted beams and the normal to the surface can be described by the following laws:

1) The incident, reflected and refracted rays, and the normal to the surface all lie in the same plane.

2) The angle of reflection is equal to the angle of incidence for all colors and any pair of substances.

3) For monochromatic (only one color present) light and a given pair of substances, a and b, the ratio of sine of the angle of incidence to the sine of the angle of refraction is a constant called the refractive index.

Figure 1 Reflection and Refraction at a Plane Surface

These basic properties of light (and all electromagnetic radiation) enable the formation of images by refraction from curved surfaces. In the first part of this experiment, you will study the properties of thin convergent and divergent lenses and their combinations. In the latter part of the experiment you will make some studies of the human eye and the way its optical performance can be modified by the use of lenses. Spectacles as well as optical instruments, such as the micropscope and telescope, are examples of such modifications.

FORMATION OF IMAGES BY LENSES AND THE FUNCTION OF THE HUMAN EYE

Summary of Thin Lens Theory

A. Convergent Lens

Figure 2 Real Image formation by a Convergent Lens

A convergent lens has two foci, F₁ and F₂, located on its axis, whose common distance form the lens center is called the focal length, f(>0) of the lens. There are 3 principal rays.

1. Parallel Ray: A ray parallel to the axis on the incident side passes through the focus on the other side.
2. Focal Ray: A ray through the focus on the incident side, emerges parallel.
3. Center Ray: A ray directed towards the center of the lens on the incident side emerges undeflected. These three rules allow geometrical construction of the image formed of any object (see Fig. 2).

We assume the object to be on the left of the lens and denote its distance from the lens by u(>0). The distance of the image from the lens is denoted by v. v is taken as positive if the image is to the right of the lens and negative if to the left. One can then prove the so-called lens formula connecting u, v, and f:

+ = (f > 0) .

The magnification , i.e. ratio of image to object size, is

m = .

Here a negative m denotes an inverted image, a positive m an erect one.

If the rays originating in an object point actually converge on an image point, so that they could be received on a screen, the image is called real. (See Fig. 2) If the rays do not actually converge but appear to come from the image point, the image is called virtual. (see Fig. 3)

Figure 3 Virtual Image Formation by Convergent Lens

B. Divergent Lens

Figure 4 Virtual Image formation by Divergent Lens

A divergent lens has two foci, F₁ and F₂, located on its axis. They have a common distance from its center. The focal length f is defined as the negative of this distance. The three principal rays are as follows (see Fig. 4):

1. Parallel Ray: A ray parallel to the axis incident from the left side, emerges as thought it was coming from the focus F₁.
2. Focal Ray: A ray incident form the left heading for focus F₂ emerges parallel.
3. Central Ray: A ray incident towards the center emerges undeflected.

The lens formula is again Equation 1.

+ = (f < 0)

but with f negative. Eq. (2) and the discussion of magnification m and of real and virtual images remain unchanged.

C. Combinations of Lenses

When two thin lenses, of focal lengths f₁ and f₂, are put close together, the combination acts like a single lens of focal length f, given by

= + .

In Eq. (4), all focal lengths are quantities with sign.

D. Power

The shorter the focal length of a lens, the greater the deflection of incident rays. Optometrists, therefore, work with the so called power of a lens, defined as

P = = .

The unit is called the diopter ( = (meter)^-1). e.g., a divergent lens with f = -50 cm has a power P = -2 diopters.

Eq. (4) for a combination of lenses can be rewritten simply in terms of power as

P = P₁ + P₂ .

THE HUMAN EYE

General Description

For the purpose of this experiment it is sufficient to consider the following model of the eye shown in Fig. 5.

Figure 5

For a normal eye, when the eye muscles are relaxed, the focal length f of the lens L has its maximum value, f^max = 2.4 cm. Thus, an object at infinity is focused on the retina which is a distance D = 2.4 cm behind L (see Fig. 5 and 6a). When the eye views a closer object, the eye muscles produce a shortening of f (so-called accommodation ) down to a minimum value f^min. The closest point on which the eye can focus is called the near point. Its distance from L, u_near, is about 25 cm for the normal eye.

Presumably, your eyes (including correcting glasses, if you wear any) are fairly close to normal.

The retina, on which are situated the light-sensitive elements (rods and cones), extends roughly over the posterior hemisphere of the eye. There are two special spots on it: The fovea, about 0.025 cm in diameter, where visual sensitivity is especially acute; and the blind spot, the area where the optic nerve is attached and which is devoid of rods and cones.

For a short-sighted (myopic) eye, the focal length f of the lens is too short. Hence, when the eye is relaxed, an object at infinity is focused in front of the retina (see Fig. 6b). The distance of the farthest point which can be found, u_far, is finite and u_near < 25 cm. A divergent lens is used for correction.

For a far-sighted (hypermetropic) eye, the focal length, f, is too long. Hence, when the eye is relaxed, an object at infinity is focused behind the retina. (See Fig. 6c). The farsighted eye can focus on objects with u_near < u < where u_near > 25 cm. A convergent lens is used for correction.

Figure 6 Normal, Shortsighted and Farsighted Eyes in Relaxed State (f = f^max).

There are two other common deficiencies of the eye. One is reduced power of accommodation, i.e. reduced variability of the focal length, f. This occurs inevitably with advancing age. The other is astigmatism in which the curvature of the front surface of the eye (the cornea) is different in different planes.