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*From*: Bernard Cleyet <bernardcleyet@redshift.com>*Date*: Wed, 9 Apr 2008 21:53:16 -0700

Wikipedia says both A and B must be true .

http://en.wikipedia.org/wiki/Paraxial_approximation

This agrees w/ the etymology and bc's view (joke)

http://www.thefreedictionary.com/para-

This (both A & B) also agrees w/ Smith Modern Optical engineering, p. 17

The military standardization HdBch (optical design) classifies rays as general, meridional, skew, and paraxial. A para. is a special type of meridional ray close to the optic axis -- I'm too lazy to continue w/ this HdBch and continue w/ the IR (military) HdBch.

It defines paraxial as first order rays (no aberrations?). Also discusses: paraxial: focal plane, focus, marginal, and principle rays. Again I'm not going to excerpt -- my impression is both must be true in order to satisfy no aberrations for both mirrors and lenses.

Also written in a std. UK text (Ditchburn, p. 240; intermediate level):

------------------------

When a spherical wave is incident upon a coaxial optical system the emergent wave is a smooth surface, though it is not, in general, exactly spherical. The rays do not meet exactly in a point. It is convenient to start by considering the behaviour of the parts of the wavefront which are near the axis, i.e. to consider rays which make small angles with the axis. These are known as paraxial rays. Since the emergent wavefront is smooth, a sufficiently small region near the axis can be replaced by a tangent sphere of equal curvature (fig. 8.13). The wave normals of this sphere meet in the point P', which is said to be the paraxial ray* image of P.

[small type paragraph:] We start by considering the rays near the axis (and the associated part of the wavefront) because these give "perfect" geometrical images. ............

*The first systematic and reasonably complete discussion of the formation of images by paraxial rays is due to C. F. Gauss ..... This discussion is, therefore, sometimes called Gaussian Optics.

--------------------

So if the image is "perfect" (w/ out aberrations) it was produced by paraxial rays, nicht wahr?

bc answered the question or lacks understanding.

p.s.

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!

We all know about some intro- texts, and you've now given a good example of those.

and:

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)?

Maybe not, Levi (Applied Optics) writes:

----------

"The formulas (sic.) derived for a thin lens may be applied to the paraxial region of a curved mirror. ..... "

------------

But these formulae were derived using paraxial rays defined as first order approximation ones. i.e. Rays in the immediate neighborhood of the optical axis, where alpha (angle WRT the principal axis) and theta (angle WRT norma at the incidence point) => 0. so sin (alpha) ~= alpha and sin (theta) ~= theta. This IIUC, is the same as A & B.

On 2008, Apr 09, , at 12:35, 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)

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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**References**:**[Phys-l] paraxial approximation***From:*Carl Mungan <mungan@usna.edu>

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