The job of the camera lens is to project an image of the outside world onto the surface of film or digital sensor. This is accomplished by altering the direction of incoming light rays. This action is called refection (Latin bend inward). In other words, light from the subject enters the lens. Refection occurs and the light rays bend inward, the revised path of the light traces out the shape of a cone. The focal length is a measurement made from the center of the lens to the apex of this cone when the lens is imaging a far distant object. If the lens is imaging a nearby object, the distance lens to cone apex is elongated. This elongated distance is now called the “back focus”. Again, the focal length is that measurement taken when the lens is imaging an object at infinity (as far as the eye can see).
The fact that focal length is a measurement from the center of the lens to the apex of the cone of image forming rays is true only for a simple lens. Camera lenses are not simple. They are complex arrays of lenses. Some will have positive power, some negative power. Some will be cemented together; some will be air-spaced. Such a complex array is necessary to mitigate lens defects called aberrations.
The fact that camera lens is complex shifts the optical center. In other words, the point associated with the lens that we measure from is not likely the center of the lens barrel. Actually there are two cardinal measuring points. Front nodal point is used to measure subject distance from the lens. Rear nodal point is used to measure focal length. These are difficult to locate; we use an optical bench setup.
A key point, we measure from the rear nodal to apex of cone of light to find the focal length, and this is performed when the lens is imaging an object located at infinity. If the object is closer than infinity, the back focus distance is longer than the focal length. This fact is why we must move the lens away from film or sensor to obtain focus when imaging an object not at infinity. This act is called focusing.
When we focus, a sharp image will be obtained if the apex of this cone of image forming rays just kisses the surface of the film or digital image sensor. Such a kiss images the cone as a tiny circle of illumination. If the distance lens to sensor (or film) is wrong, this circle will be larger. Sharpness is obtain only when this circle is so small that it will be perceived as a point of light, a point has no desirable dimension. As an example, newspaper photos appear un-sharp because the dots of ink that comprise the image are too big, they are desirable.
“Therefore my first question is this. If the focal length is the distance from the convergence point to your camera sensor and since the image is the sharpest where the refracted beams of light meet how is the image not blurry since the location of the "intersection" is different from the location of the sensor.”
Answer: We focus by moving the lens forward and backward. This act adjusts the position of the apex (intersection) so that it just kisses the surface of the sensor. This act results in a tiny circle of illumination being recorded. This circle is called the “circle of confusion” because under the microscope it is seen with scalloped edges and it is juxtaposed with all the other circles of confusion that make up the image. To appear as sharp, this circle, the smallest fraction of an image, must be 0.5mm in diameter or less when viewed from standard reading distance.
In the SLR design, the image forming rays are diverted for viewing and composing via a mirror and pentaprism. The falls on a roughed up glass screen (ground glass viewing screen). The distance rear nodal to sensor and rear nodal to viewing screen are carefully maintained so they are identical.
“Secondly how is technically and inside the camera the refraction amount by the lens translated to a wide angle or telephoto lens.”
Answer: When we fit lenses with a focal length that is approximately equal to the corner to corner measurement of the sensor, the angle of view will be classified as “normal”. This will be about 45° when the camera is held horizontal (landscape). If the focal length is shorter, say 70% of this value, or shorter, the angle of view is classified as wide-angle. If the focal length is 2X this value, or longer, the lens will be classified as telephoto.
So far, we are describing how light rays from an object on optical center preform. The film/sensor is being bombarded by billions and billions of rays from everything within its field of view. Points emanating from all these objects, each trace out an image cone of light. Each will have a different length depending on subject distance. We stop down the lens using a restriction called an aperture. This reduces the working diameter of the lens.
Such a restriction reduces the diameter of the base of image cone. This act of stopping down , yields a skinny image forming cone of light. Imagine two sharpened pencils pointy end to pointy end. The narrower the pencils, the more tolerance fore and aft of the apex and we still get a tiny circle. All we need is 0.5mm.
The rub is, today’s miniature cameras yield an image too tiny to be useful. We must enlarge them perhaps 8X or even 12X to get an 8X10 inch image. Thus the circle size at the sensor must be super tiny to support this blowup. Industry stand for good focus is a circle 1/1000 of the focal length. Thus for a 50mm lens the circle size at the image plane is 50mm ÷ 1000 = 0.05mm. For precision work we often use 1/1500 of focal length, that’s 0.033mm. Did you ever dream of such exactness?