Re: [Phys-l] paraxial approximation

[John Denker <jsd@av8n.com>]

On 04/09/2008 12:35 PM, Carl Mungan wrote:
....
> I hope you see where I'm going with this....

> and yet one will get spherical aberrations ....

Well, after going back and re-reading the original questions, I'm
less sure than ever where Carl is going with this, but I suspect
there is more to the question than previous answers (including 
mine) have dealt with.

Is this about the terminology, or about the ideas???  I can never
get too excited about terminology.  Terminology is only important
insofar as it helps us formulate and communicate the ideas.  If
the terminology starts getting in the way, it's time to come up
with better terminology.

In the present case, it may help to *extend* the idea of "paraxial
approximation" in analogy to the Born approximation.  Conventionally
when people speak of _the_ Born approximation, they mean the _first_
Born approximation.  Meanwhile the 2nd Born approximation, 3rd
Born approximation, etc. continue to exist.

When we use this to talk about optics, we can say:
 *) To zeroth order, every lens and every mirror is /flat/ when we 
  hit it on-axis.  This is the trivial approximation.
 *) To second order, every lens and every mirror is /parabolic/.
  Spheres are indistinguishable from paraboloids.  This is commonly
  called "the" paraxial approximation.  From our new viewpoint we
  see it as the lowest-order nontrivial paraxial approximation.
 *) Odd-order terms vanish by symmetry.
 *) If we are interested in aberrations, which we sometimes are
  and sometimes aren't, we can look at the fourth-order terms.
  This allows us to quantify the difference between a sphere 
  and a paraboloid.  From our new viewpoint we see this as a 
  /higher-order/ paraxial approximation.
 *) And so on to higher orders if necessary.  Maksutov corrector
  plates et cetera.


The idea is to think of it as a series, and to keep "enough" terms
in the series to suit the problem at hand, whatever that may be.

The idea is clear, even if the conventional terminology is not.

=================

The paraxial approximation often gets tangled up with the thin-lens 
approximation.  I'm not sure it makes sense to separate them, except
in special cases.  When a lens is thin, the only thing that matters 
is the position where the ray hits the lens, and the figure of the 
lens;  in particular you can do whatever you want with the angle.

In contrast, whenever something has thickness -- even a thick slab
of free space -- you need to worry about the angle.

In phase space, lenses and spacers are complimentary:
 -- A lens changes the angle in proportion to the off-axis distance.
 -- A spacer changes the off-axis distance in proportion to the angle.


For a lens, you can at least imagine a super-high index of refraction
such that the lens is thin and flat yet still has a short focal length.
For mirrors, you cannot even imagine that;  the mirror cannot be thin
and flat and still have a short focal length.