# Geometrical construction of images

Images can be constructed by using simple optical and geometrical rules. As can be seen on the following drawing, at least two rays are used to create the image of an object.

There are three basic rules to follow:

• Objects taken at various distances touch the optical axis with one end.
• By definition, rays that pass through the center of the lens do not change direction, that is, in the center, a lens behaves like parallel glass and no refraction occurs.
• By definition, all rays parallel to the optical axis pass through the focus.

There is a very basic lens formula, worth mentioning, which we use when calculating the light falling onto a CCD chip:

1/D + 1/d = 1/f

where D is the distance from the object to the lens, d is from the lens to the image, and f is the focal length of the lens. Note that d refers to a “non-infinite” distance object image and that is why it is bigger than f, whereas if the object is at an innite distance, d would be equal to f.

Please note the position of images for various distance objects. Lens focusing is achieved by changing the distance between the lens and the image plane (which is where the CCD/CMOS chip is located). So, only when a lens is focused at an infinitely far object does the image projection coincide with the focus plane. In all other cases the distance between the lens and the image is bigger than the focal length of the lens (the lens is pushed away from the imaging chip).

It should also be noted that in practice, a lens is composed (as discussed earlier) of many optical elements. Therefore, they are represented by an equivalent single-element lens located at the principal point.

A lens composed of many optical elements (single thin lens) has two principal points called primary and secondary principal points. For a thin lens, these points coincide and they are located at the center of the lens.

The planes that pass through these principal points and are perpendicular to the optical axis are called principal planes.

The principal planes have the following properties:

• A ray incident to the primary principal plane (and parallel to the optical axis) will leave the secondary principal plane at the same height, traveling toward the focal point (focus).
• An incident ray directed toward the primary principal point will leave the secondary principal point at the same angle.
• The focal length of such a lens is measured from the secondary principal plane to the focus.

Using the above properties, we can construct a geometrical image in the same manner as was shown with the single optical element.

The secondary principal point may fall outside the group of lenses. This is the case with very short focal length lenses. The shorter the focal length is, the more optical elements have to be added for correcting various distortions, making the lens more expensive. With the CCD chip reduction (2/3” down to 1/2”, then to 1/3”, and now to 1/4”), shorter focal length lenses have to be manufactured in order to preserve the same wide angle as the preceding chip sizes. This, in turn, has forced the industry to reduce the C-mount 17.5-mm back-flange distance in order for the optics to get simpler, smaller, and cheaper. The new format of back-flange distance is 12.5 mm, and since it is smaller, it is referred to as the CS-mount standard.

## Aspherical lenses

As mentioned earlier, spherical aberration is a common distortion that appears in the majority of lenses of a spherical type. Spherical-type lenses are the most common since they are produced by grinding and polishing in the easiest mechanical way, following the spherical laws. This refers to a circular machine polishing with the result being a lens of a spherical appearance. It can be shown that apart from the chromatic aberrations present in a single-lens element (the “color decomposition” of white light), aberration also occurs because of the spherical profole of the lens. The focus is not a very precise single point.

Theoretically, using the physical laws of refraction, we can show (but we will not go into the details) that a bell-shaped lens, which does not follow the spherical law, is the ideal shape for obtaining a single focusing point without spherical distortions. The cross-section prole of such a lens is a curve that deviates slightly from a circular shape, appearing more bell shaped. This type of lens is called an aspherical lens. The drawing on the next page shows this in an exaggerated form in order to help the reader understand it.

Understandably, such a shape is hard to produce by regular polishing techniques, but, if properly manufactured, it offers quite a few advantages over the conventional spherical lenses, including higher iris openings (which is reected in a lower F-stop), wider angles of view, shorter minimum object distances, and fewer optical elements because there are fewer aberrations to correct (thus resulting in lighter and smaller lens designs).

This technology is more expensive due to the aforementioned complex polishing techniques

Many optical companies are now manufacturing molded aspherical lenses, avoiding the critical process of grinding. This process does not offer the same glass quality as the regular one, but it does offer a solution for more economical production of aspherical lenses. The quality of such lenses is yet to be proven, but they do exist and are available in the CCTV market as well

## CTF and MTF

What we want from a lens is sharp and clear images, free of distortions.
As already mentioned, lenses have limited resolving power, and this is especially important to have in mind when using them in high-resolution systems, such as high denition and mega pixel cameras.

Resolution refers to the lens’s ability to reproduce fine details. In order to measure this ability, a chart that consists of black and white stripes with various density (spatial periods) is used. This is usually expressed in lines per millimeter (lines/mm). When counting how many lines/mm a lens can
resolve, we count both black and white lines.

A characteristic that shows the “response” of a lens to various densities of lines/mm is called a Contrast Transfer Function (CTF)

Theoretically, it is better to know the lens characteristics for a continuous variation of black to white (in the form of a sine wave), and not just for stripes that abruptly change from black to white. This would be especially suitable for TV lenses since the optical signal is converted into an electrical signal with which sine waves are easier to represent and evaluate. This characteristic is known as a Modulation Transfer Function (MTF).

In practice, however, it is much easier to produce a test chart with just black/white stripes rather than the sine wave variation between black and white. CTF is not exactly the same as MTF, but it is much easier to measure and is good enough to describe the lens’s global characteristics.

The easiest analogy of MTF to understand would be the spectral response of an audio system. In an audio system we usually describe the output level (voltage or sound pressure) versus the audio frequency. In optics it is similar, where MTF is expressed in contrast values (from 0 to 100%) versus spatial frequency (expressed in lines/mm).

This site uses Akismet to reduce spam. Learn how your comment data is processed.