Talk:Paraxial approximation

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[edit] Order of approximations

When one talks about the order of an approximation, one is talking about the largest power of the variable to be included. \cos(\theta) \approx 1 - { \theta^2 \over 2 } is a second-order approximation in θ. To first order, \cos(\theta) \approx 1 . Thus, for cos(θ), the first order and zeroth order approximations are the same. In any event, the article is about the paraxial approximation. I am pretty sure that the form you put in is not used in paraxial raytracing, which uses equations that are linear in θ.--Srleffler 12:47, 29 January 2006 (UTC)

I just completed a graduate-level course at a major university in optoelectronics. We made extensive use of the paraxial approximation throughout the course, not only in ray optics, but also in wave optics and electromagnetic optics. As you rightly pointed out, the issue is not really about the order of approximation, which is really a matter of semantics more than anything. The issue is to make sure that whatever approximation you take will lead to valid results. If you approximate cosine as 1, and ignore the second term of the Taylor series, I believe that you will end up with incorrect results in certain situations. If, on the other hand, you include the first two terms of the Taylor series, you should end up with good results as long as the angle is "small." Besides, including the "extra" term does no harm, but ignoring it can lead to bad results. -- Metacomet 15:57, 29 January 2006 (UTC)
I'll have to defer a detailed reply until I have my optics texts handy. Note, though, that the issue here is not whether one approximation or another will lead to valid results, but only which approximation is actually used as the paraxial approximation. See WP:NOR if you do not see the distinction I am making. Our role as editors of this article is to document what people who use the paraxial approximation do, not to figure out what approximation is best.
You may be right, or perhaps both are used in different situations. I'm pretty sure that paraxial raytracing does not include any θ2 terms, but of course the "paraxial approximation" is used in other areas of optics, as you noted. --Srleffler 19:07, 29 January 2006 (UTC)
I wrote the above before I saw your latest edit to the article. Looks like we were thinking along the same lines. I changed the text regarding orders though. Understanding orders of approximation is important, for just the reason you mentioned above: if you want to get correct results, you need to use the right approximations. If you approximate different things to different orders in a single calculation, terms that should cancel may fail to do so, giving a less accurate result. It gets confusing with the trig functions, since the even-ordered terms of sine and tangent and the odd-ordered terms of cosine all have zero coefficients. The difference between the two approximations for cosine amounts to approximating all three trig functions to first order or second order. --Srleffler 19:33, 29 January 2006 (UTC)

Actually, the topic ought to reach farther: There is also an angle between the ray and the plane chosen for the usual representation of rays. I believe if you ignore that angle you exclude astigmatism and coma, or parts of the astigmatic and comatic point images. There are four relevant terms: paraxial, meridional, sagittal, and skew. It needs to be sorted out and the reader referred onwards. Carrionluggage 20:18, 29 January 2006 (UTC)

As it stands right now, this article is still a stub, and a pretty early one at that. There is plenty of work to be done. I am sure we will be able to iron out the issues over time. -- Metacomet 21:24, 29 January 2006 (UTC)