02. Object distance, image distance, and magnification
Description
This article is from the Photographic Lenses Tutorial, by David Jacobson with contributions by others.
02. Object distance, image distance, and magnification
Throughout this article we use the word "object" to mean the thing of which an image is being made. It is loosely equivalent to the word "subject" as used by photographers.
In lens formulas it is convenient to measure distances from a set of points called "principal points". There are two of them, one for the front of the lens and one for the rear, more properly called the primary principal point and the secondary principal point. While most lens formulas expect the object distance to be measured from the front principal point, most focusing scales are calibrated to read the distance from the object to the film plane. So you can't use the distance on your focusing scale in most calculations, unless you only need an approximate distance. Another interpretation of principal points is that a (probably virtual) object at the primary principal point formed by light entering from the front will appear from the rear to form a (probably virtual) erect image at the secondary principal point with magnification exactly one.
"Nodal points" are the two points such that a light ray entering the front of the lens and headed straight toward the front nodal point will emerge going straight away from the rear nodal point at exactly the same angle to the lens's axis as the entering ray had. The nodal points are identical to the principal points when the front and rear media are the same, e.g. air, so for most practical purposes the terms can be used interchangeably.
In simple double convex lenses the two principal points are somewhere inside the lens (actually 1/n-th the way from the surface to the center, where n is the index of refraction), but in a complex lens they can be almost anywhere, including outside the lens, or with the rear principal point in front of the front principal point. In a lens with elements that are fixed relative to each other, the principal points are fixed relative to the glass. In zoom or internal focusing lenses the principal points generally move relative to the glass and each other when zooming or focusing.
When a camera lens is focused at infinity, the rear principal point is exactly one focal length in front of the film. To find the front principal point, take the lens off the camera and let light from a distant object pass through it "backwards". Find the point where the image is formed, and measure toward the lens one focal length. With some lenses, particularly ultra wides, you can't do this, since the image is not formed in front of the front element. (This all assumes that you know the focal length. I suppose you can trust the manufacturer's numbers enough for educational purposes.)
So object to front principal point distance. Si rear principal point to image distance f focal length M magnification
1/So + 1/Si = 1/f M = Si/So (So-f)*(Si-f) = f^2 M = f/(So-f) = (Si-f)/f
If we interpret Si-f as the "extension" of the lens beyond infinity focus, then we see that this extension is inversely proportional to a similar "extension" of the object.
For rays close to and nearly parallel to the axis (these are called "paraxial" rays) we can approximately model most lenses with just two planes perpendicular to the optic axis and located at the principal points. "Nearly parallel" means that for the angles involved, theta ~= sin(theta) ~= tan(theta). ("~=" means approximately equal.) These planes are called principal planes.
The light can be thought of as proceeding to the front principal plane, then jumping to a point in the rear principal plane exactly the same displacement from the axis and simultaneously being refracted (bent). The angle of refraction is proportional the distance from the center at which the ray strikes the plane and inversely proportional to the focal length of the lens. (The "front principal plane" is the one associated with the front of the lens. It could be behind the rear principal plane.)
Beyond the conditions of the paraxial approximation, the principle "planes" are not really planes but surfaces of rotation. For a lens that is free of coma (one of the classical aberrations) the principle planes, which could more appropraitely be called equivalent refracting surfaces, are spherical sections centered around and object and image point for which the lens is designed.
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