Re: [Phys-l] paraxial approximation

["Bob Sciamanda" <trebor@winbeam.com>]

Carl,
The typical development of the paraxial approximation involves approximating 
sines and cosines by their first order expansions.  This includes not only 
the angle which a ray makes with the optical axis, but also the angle which 
a ray makes with the normal to the spherical refracting/refracting surface. 
This latter angle can be large for a ray parallel to the axis depending on 
where it strikes the spherical refracting/reflecting surface.
(Refer to the development in any optics text)

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
trebor@winbeam.com
http://www.winbeam.com/~trebor/
----- Original Message ----- 
From: "Carl Mungan" <mungan@usna.edu>
To: <phys-l@carnot.physics.buffalo.edu>
Sent: Wednesday, April 09, 2008 3:35 PM
Subject: [Phys-l] paraxial approximation


> 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)
>
> WHICH OF THE THREE CHOICES (1) TO (3) IS CORRECT?
>
> Before answering, consider a second, application question. (After
> all, one can make any definition one likes. Without an application in
> mind, it just becomes semantics instead of physics.)
>
> QUESTION II. If the paraxial approximation is valid for a spherical
> mirror, then one can state:
> (4) there will be no aberrations of any kind
> (5) some kinds of aberration will be eliminated, but other kinds will 
> remain
> (6) the paraxial approximation isn't very useful/relevant/interesting
> when applied to spherical mirrors; it's primary application is to
> other shapes (perhaps parabolic, elliptical, etc)
>
> WHICH OF THE THREE CHOICES (4) TO (6) IS CORRECT?
>
> I hope you see where I'm going with this. My reading of typical intro
> textbooks is that their answers are (1) and (4). (Check out the intro
> text *you* use to see.) But these two answers are inconsistent with
> each other! For example, a collimated beam can be incident [so that
> every ray is parallel to the principal axis and (A) certainly holds]
> and yet one will get spherical aberrations if some incident rays are
> far off-axis. I believe some texts try to avoid this by instead
> choosing answer (3). But now I'm wondering whether that's overkill.
> Wouldn't choice (2) avoid spherical aberrations? One might think
> you'd get coma, but I'm not sure that's correct. It's true the focal
> surface is curved, but provided I use a mirror of small diameter (or
> equivalently aperture it right near it), won't an off-axis object
> give a reasonably sharp image nonetheless (albeit located off-axis)?
>
> --
> Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
> Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
> mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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