Mathematics of Human Vision and Corrective Lenses

Mathematics of Human Vision and Corrective Lenses

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It is a daily ritual...wake up, eat, shower, contacts, class. I do this every morning, however I have never stopped to think, "Why?"
No, not "Why am I going to class so early?" but "Why do I need these contacts?" With this project I will attempt to explain the basics of vision and corrective lenses, as I, myself, finally learn the reason for sticking my finger in my eye every morning.

Basic Eye Anatomy

The cornea is responsible for protecting the eye and for refracting incoming light rays.

The pupil is merely an opening that allows light to enter into the eye. Its black color is attributed to the fact that light is not able to exit the eye through the pupil.

The iris acts to control the size of the pupil. In bright light, the iris is dilated in such a way as to reduce the size of the pupil and limit the amount of entering light. In dim light, the iris adjusts its size as to maximize the size of the pupil and increase the amount of incoming light.

The crystalline lens is a fibrous, jelly-like material that serves to fine tune the vision process by adjusting its shape and therefore the focal length of the system.

The ciliary muscles relax and contract to change the shape of the lens.

The retina contains rods and cones which detect the intensity and frequency of incoming light and, in turn, send nerve impulses to the brain.

Behind the Eye

The four main components of the eye that are responsible for producing an image are the cornea, lens, ciliary muscles and retina. Incoming light rays first encounter the cornea. The bulging shape of the cornea causes it to refract light similar to a convex lens. Because of the great difference in optical density between the air and the corneal material and because of the shape of the cornea, most of the refraction to incoming light rays takes place here. Light rays then pass through the pupil, and then onto the lens. A small amount of additional refraction takes place here as the light rays are "fine tuned" so that they focus on the retina.

This is a representation of the eye's lens system. This eye has no eye condition, such as nearsightedness or farsightedness, and the lens is drawn in its relaxed position. The light rays are focused appropriately on the retina. The thickness of the cornea is 0.449 mm, the distance from the cornea to the lens is 2.

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794 mm, and the thickness of the lens is 4.979 mm. The front radius of the cornea is 7.259 mm and the back radius is 5.685 mm. The front radius of the lens is 8.672 mm and the back radius is 6.328 mm. The distance from the front of the cornea to the retina is 24.0 mm. The path of light rays was traced using the indexes of refraction shown above and these measurements.

Vision is dependent on the stimulation of nerve impulses by an incoming light photon, which only real images are capable of producing. The resulting image produced on the retina is formed by the actual convergence of light rays at a point in space and is, therefore, a real image. The resulting image is also inverted and reduced in size, as to allow the entire image to fit on the retina. The brain compensates for the inversion and interprets the signal as originating from a right side up object and produces the "flipped" image that we see.

Accomodation

A small area of the retina, known as the fovea centralis, with a diameter of approximately 0.25mm contains the greatest concentration of rods and cones, making it the optimal location for the formation of an image. The eye rotates in its socket in order for images to be focused at this location. The distance from the cornea to this small area is approximately 1.7cm, so light entering the cornea must produce an image with a fixed image distance of 1.7cm. To accommodate this, the eye must be able to alter its focal length in order to focus images of objects, either nearby or far away, on the retinal surface. This is known as accommodation. The following tables illustrate the importance of this process. The calculations were made using the lens equation.

Table 1
Image distances of an object at varying distances
from the eye using a fixed focal length of 1.7cm

Table 2
Dependence of focal length on varying object
distances with the image distance fixed at 1.7cm

Objects located at varying distances from a lens system with a fixed focal length produce images which are at varying distances from the lens (Table1). However, the eye must always produce images on the retina, which is always a fixed distance from the cornea. In order to focus images with a fixed image distance the eye must accommodate, and alter its focal length (Table2).

Nearby objects are focused at further distances than far away objects. The eye accommodates for this by having the ciliary muscles contract and squeeze the lens into a more convex shape, increasing the curvature of the lens. This causes more refraction resulting in a shorter focal length and an image that is brought back closer to the cornea and upon the retina. (Figure 1)

Far away objects are focused at a closer distance than nearby objects. The ciliary muscles therefore relax to reduce the convexity of the lens, decreasing the curvature and decreasing the refraction so that the focal length is longer and images are pushed away from the cornea and onto the retina. (Figure 3)

Conditions and Corrections

The eye's ability to accommodate allows it to view focused images of both nearby and far away objects. The inability to provide a large variance in focal lengths leads to vision defects.

Myopia (Nearsightedness)
Nearsightedness is the inability of the eye to focus on distant objects. This condition is most common in youth and can be attributed to a bulging cornea or elongated eyeball. Light from distant objects is refracted more than necessary with a bulging cornea, as the convexity is greater, and the image is formed at a location in front of the retina. If the eyeball is elongated horizontally then the retina is placed at a further distance from the cornea-lens system and images of distant objects form in front of the retina. In either case, the image is not focused on the retina and this results in a blurry image of distant objects.

To correct this condition, a diverging lens is used. The total refracting power of the eye is reduced when the light diverges before reaching the eye and then is converged by the cornea and lens to produce an image on the retina.
Hyperopia (Farsightedness)
Farsightedness is the inability to focus on nearby objects. This condition usually presents itself later on in life, when the ciliary muscles weaken and the flexibility of the lens is decreased. The lens can then no longer assume the highly convex shape, that is needed to focus nearby objects. The total refracting power of the eye has diminished and images are focused behind the retina. If this condition occurs in younger people, the cause is most likely because of a small, or short, eyeball. The retina then lies closer than usual to the cornea-lens system and images of nearby objects are focused behind the retina. In both cases, the image is not focused on the retina and a blurry image results.

To correct this condition, a converging lens is used. This assists the eye's lens by refracting light before it enters the eye so that the image distance is decreased and the image of nearby objects is once again focused on the retina.

Corrective Lenses

Corrective Lenses can either be spherical or cylindrical. The purpose of an eyeglass is to alter the focal length and point of focus in the eye. The focusing ability of an eyeglass is determined by the difference in curvature on its front and back surface, its thickness and its index of refraction. The precise lens can then be chosen to correct any vision condition.
Spherical Lenses

* Diverging (Concave)

A diverging lens diverges rays of light which are travelling parallel to its principal axis. These lenses are thinner across their middle and thicker at their upper and lower edges. Diverging rays can be traced backwards until they intersect in a point, as seen below for a double concave lens. This is the focal point, F.

* Converging (Convex)

A converging lens converges rays of light which are travelling parallel to its principal axis. These lenses are thicker across their middle and thinner at their upper and lower edges. Converging rays can be traced until they intersect in a point, as seen below for a double convex lens. This is the focal point, F.

Cylindrical lenses are used most often to correct astigmatism.

Aberrations and the Eye

Aberrations are errors in an image that occur because of imperfections in the optical system. There are many different kinds of aberrations but here we will look at only two that are present in the eye.

* Spherical Aberration
These occur because rays entering the eye closer to the center of the pupil are focused behind rays entering further away from the optical axis. The degree of variation depends on the amount of accommodation.


Spherical aberration with a general, spherical lens. Rays leaving the same point are focused at different locations, depending on their distance from the optical axis.

* Chromatic Aberration
These occur because the eyes index of refraction is greater for shorter wavelengths than for longer wavelengths, therefore blue light rays are focused in front of red light rays.


This is an exaggeration of chromatic aberration for a general, spherical lens. Rays coming in parallel to the lens are focused at different locations, depending on their wavelengths.

These problems are often compensated for by neurological processing and present themselves only in certain situations, such as night work, where the pupil is dilated and outer parts of the eye contribute to spherical aberration.

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