First, a convex lens is one which is thicker in the center than it is near the edges. This is shown in this diagram:
When we look at the cross-section of a convex lens we notice that the edges resemble prisms. In fact, a stack of prisms of varying angles can be used to simulate the actions of a convex lens. One such is shown here and is called a Fresnel Lens.
Light passing through the angled prisms near the edges is bent significantly while light passing through the flat, central area is hardly bent at all. If the angles are calculated correctly, light rays which are parallel to one another when approaching such an arrangement can be brought together in a small area as illustrated here:
The area where the light rays are converged is called the focal area.
Now with convex lenses, the sides are continuously curved allowing the light to be focused into a point rather than a larger area. This would be called the focal point.
DEFINITION 1:
The focal point of a convex lens is the point where light rays parallel to the axis are brought to a point. The distance from the lens to this point is called the focal length of the lens.
Because it seems rather odd to represent light as a dark line on a white page, the diagram above has been inverted below to show white light on a black background. The principle is the same.
Now the question is where one would find parallel light rays in nature? How common or uncommon are parallel light rays if most of the light we seen on a daily basis is diverging to one degree or another?
If an object is very far away, the angle formed between adjacent light rays is very small. Depending on the focal length of the specific lens, this distance might be anywhere from a few meters to a kilometer. If the object is very far, say 93,000,000 miles (1.5 x 1011 m) like the Sun, the distance is sufficiently far that light rays are essentially parallel. So sunlight is a convenient source of parallel light rays. Objects that are a great distance away like hills or trees may also furnish almost parallel rays. Finally, lasers are a relatively inexpensive source of parallel light due to their inherent nature.
NOTE: The light rays do not stop when they get to the focal point. They just happen to pass through this point and continue their journey on into the universe.
NOTE 2: In the diagrams above, light rays are shown bending at the center of the lens. This is a construction technique and is used only for convenience. In fact the rays would bend once upon entering the lens and a second time upon exiting.
BOTTOM LINE: If we see a light ray that's parallel to the axis of a convex lens we know where it is going to go on the other side -- through the focal point.
DEFINITION 2:
Diverging light rays striking a convex lens can be bent until they emerge parallel to the axis. The point where this happens is called the focal point.
Or as before, white light on a black background:
NOTE: Because we have defined "focal point" so precisely, we can understand that a light ray that is not parallel to the axis will not pass through the focal point on the other side of the lens. Also we know that a light ray that does not pass through the focal point will not emerge parallel to the axis on the other side.
BOTTOM LINE: If we have a light ray that either starts at the focal point, passes through the focal point or looks to the lens like it starts at the focal point, that light ray will be bent until it is parallel to the axis.
BIG NOTE:
A convex lens has two focal points - one on each side. They are equal distances from the lens. The lens does not have to have the same curvature on both sides for this to be true, and it doesn't depend on the direction the light takes entering the lens. It is the combined curvature that determines the focal point.
APPLICATIONS:
In an astronomical telescope, we focus the light coming from a distant stars onto a piece of photographic emulsion. Using a convex lens and placing the emulsion one focal length away will accomplish this task. The light coming from distant stars is very parallel.
If we wish to concentrate the light coming from the sun onto a small area we might choose a convex lens. The light could be focused onto a small photoelectric unit which would generate electricity for us. Making large lenses might be cheaper than making photoelectric cells.
If we wished to send out a beam of parallel light, we could place a small light source one focal length away from a convex lens. The result would be a parallel beam of light for one use or another. For example, it could be used to direct the light from a traffic signal to the lane which needs to see it rather than to all of them.