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A convex lens is a converging lens, meaning that light rays entering the lens parallel to its axis are converged as they pass through the lens. When they come out the other side of the lens they will strike the focal point behind the lens. The process by which the lens bends the light rays is called refraction.
A concave lens is a diverging lens. Light rays are refracted so that they diverge as they pass through the lens.
For either lens type the refraction is caused by the angle between the light ray and the surface of the lens. The greater the curvature of the lens, the greater the angle between the ray and lens surface (angle of incidence).
At the pole (axis) of the lens the lens surface is exactly perpendicular to the light path, representing a zero incidence angle. The ray will simply pass through the lens without refraction (bending) and continue out the lens to pass through the focal point, regardless of the distance of the focal point behind the lens. It is only the light rays that are off center from the lens’s axis that are bent. They will be angled in a way to converge on the focal point behind the lens.
These concepts are illustrated in the upper left diagram on the first page of the reference. One can see what happens to the light ray passing straight through the lens’s axis (pole), versus the rays above and below the axis that are bent by the lens as they go through.
This is because the ray is entering and leaving the lens at an angle of incidence of 0 degrees (or at a normal) to the glass/air interface at those points. For any value of refractive index the angle of refraction must be 0 degrees so the ray is undeviated.
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