# Talk:Paraxial approximation

WikiProject Physics (Rated Start-class, Mid-importance)
This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Start  This article has been rated as Start-Class on the project's quality scale.
Mid  This article has been rated as Mid-importance on the project's importance scale.

## 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)

## Clarification Requested

Regarding: meridional ray and sagittal ray

The geometrical optics problem is often symmetric about the optic axis. Many planes may be constructed containing the optic axis; only two rays are required to construct such a plane.

For this introductory article, the conventional terminology is ambiguous without chasing several other references. Please clarify how the distinction of meridional and sagittal rays relates to the accuracy of the approximation. Ryan Westafer (talk) 23:04, 7 November 2010 (UTC)

I'm not sure what the ambiguity you refer to is. As you note, many planes can be constructed which contain the optic axis. Any ray that lies in any of those planes is a meridional ray. Saggital rays do not lie in any plane that contains the optic axis.--Srleffler (talk) 23:32, 7 November 2010 (UTC)

## Range of validity

The error for the approximation is less than 1% for angles smaller than 13.9857° and less than 0.5% for angles smaller than 9.9066° Solution from wolframalpha.com — Preceding unsigned comment added by Rodolfo Hermans (talkcontribs) 11:50, 26 May 2011 (UTC)

I added a link to a live plot using Wolfram Alpha for all three approximations. The 0.5% error for <10° angles is only valid for the sine approximations, especially the cosine works in a much smaller domain, namely <5°.
Plot[{(x Deg - Sin[x Deg])/Sin[x Deg], (Tan[x Deg] - x Deg)/Tan[x Deg], (1 - Cos[x Deg])/Cos[x Deg]}, {x, 0, 15}]
2pem (talk) 21:49, 15 January 2014 (UTC)

## Angle change?

As I understand it, for an idealized finite-sized paraxial lens (e.g., a Paraxial surface in ZEMAX) a given point on the (flat) surface of a paraxial lens changes the angle of the incoming ray by a fixed angle regardless of the angle of incidence. Is that right? If so, I think that angle change is as follows: At radius r from the axis, a paraxial lens of focal length f bends a ray parallel with the axis by $\theta$ such that it hits the optical axis a distance f downstream. That ray intersects the axis with the same angle, $\theta$. Thus

$\tan \theta = \frac{r}{f}$

and so

$\theta(r) = \arctan\frac{r}{f}$

and by extension, a ray hitting the paraxial lens at radius r and coming toward the axis with angle $\varphi$ comes out at $\varphi+\theta(r)$. Does that sound right? —Ben FrantzDale (talk) 15:09, 30 November 2011 (UTC)

## NA

I don't see mention of how it changes numerical aperture. This is original research, but here's what I got: If you have a paraxial lens with focal length f a distance d from an object point hitting the lens at radius r for a marginal ray angle of $\theta_1$, the output ray will have angle $\theta_2=\theta_1 - \arctan\frac{r}{f}$ as in my post above. If you assume n is everywhere unity, then NA=sin θ. For small angles, which is already the assumption, we have

$\theta_1 = \arctan\frac{r}{d} \approx \frac{r}{d}$
$\theta_2 = \theta_1 - \arctan\frac{f}{d} \approx \frac{r}{d}-\frac{r}{f}$

and so

$\sin\theta_1 \approx \frac{r}{d}$
$\sin\theta_2 \approx \frac{r}{d}-\frac{r}{f}$

and so, assuming n=1:

$\mathrm{NA_1} \approx \frac{r}{d}$
$\mathrm{NA_2} \approx \frac{r}{d}-\frac{r}{f} = \mathrm{NA_1}-\frac{r}{f}$.

This appears to be right from a few examples in ZEMAX. —Ben FrantzDale (talk) 16:13, 30 November 2011 (UTC)

If you're trying to improve your understanding of the subject, it would be better to ask your questions on the Science Reference Desk. Article talk pages are for discussion related to improving the article.--Srleffler (talk) 02:08, 1 December 2011 (UTC)