Re: [Phys-l] paraxial approximation

[John Denker <jsd@av8n.com>]

On 04/09/2008 12:35 PM, Carl Mungan wrote:
> I am a little bit confused about the precise statement of the 
> paraxial approximation. For simplicity, let's restrict the discussion 
> to image formation by a single curved mirror. Consider the following 
> two statements:
>
> (A) optical rays make small angles relative to the principal axis
> (B) optical rays strike the mirror near the vertex V (defined as the 
> point of intersection of the principal axis with the mirror)
>
> QUESTION I. The paraxial approximation is defined to hold whenever:
> (1) rays satisfy (A) regardless of whether or not they satisfy (B)
> (2) rays satisfy (B) regardless of whether or not they satisfy (A)
> (3) rays satisfy both (A) and (B)

This question, like so many others in optics (and physics generally)
is most easily understood in phase space.
  http://www.av8n.com/physics/phase-space-thin-lens.htm

The simplifying assumptions behind the paraxial approximation
involve points that are near the origin _in phase space_.  That
means *both* the X dimension and the dX/dZ dimension.

It pretty much has to be that way.
 -- If something has a big dX/dZ, even if the X value started out
  small it won't stay small, as soon as there is any free-space
  propagation.
 -- If something has a big X, even if the dX/dZ value started out
  small it won't stay small, as soon as it hits a lens or mirror.


Geometric optics is in large measure "phase space gymnastics".  Phase
space tells you what you can do and what you can't.

This ties into many other branches of physics, including particle accelerator
design, the uncertainty principle, the 2nd law of thermodynamics, ........