Talk:Gaussian beam

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[edit] Paraxial approximation

As far as I understand it, the derivation of the Gaussian beam as a solution to the wave equation relies crucially on the assumption that the rays comprising the beam (meaning the wavefront normals) are paraxial rays, which means by definition that θ is much smaller than 1, which likewise means that θ is very nearly equal to its tangent. If this condition is not true, then the entire solution is invalid, and the beam is not Gaussian. In other words, a Gaussian beam always consists of paraxial rays.

Is that not correct? If not, please explain the issue.

Thanks.

-- Metacomet 07:08, 17 February 2006 (UTC)

Yes, I think you're probably right. I had taken my interpretation from Yariv's book, which says

This last result is a rigorous manifestation of wave diffraction according to which a wave that is confined in the transverse direction to an aperture of radius w_o will spread (diffract) in the far field...according to [the equation above].

I took this to imply that the tan θ form was accurate even in the regime where the paraxial approximation fails, but it's not clear that this is the case. Siegman uses the linear form, and also shows explicitly that the gaussian beam model fails for cases where the divergence is greater than about 30°. He gives references to papers that give higher order corrections to Gaussian beam propagation theory. Note that θ need not be much smaller than 1 rad according to Siegman's analysis, though. --Srleffler 23:22, 17 February 2006 (UTC)

[edit] Notation

I reverted Metacomet's reversion of Pmg's notation change. I don't intend to start an edit war on this, but I do prefer the small-w notation (as apparently does Pmg), and the lowercase notation is used in the figure on this page. This article used this notation until about six weeks ago, when Metacomet changed it from w to W, while doing a major upgrade to the contents of the page. I objected at the time, but he disagreed and I didn't pursue the matter. I understand his/her concern that w is (in some fonts) too easily confused with ω. The lowercase notation is used in Yariv's book and, I think, in Siegman's. The uppercase is apparently used in Saleh and Teich. Obviously we need to arrive at some consensus on notation. I'm not sure how best to do that in this case. Anybody have any thoughts on this issue? The confusion between beam size and angular frequency is not a big problem. I actually used a lower-case omega (which in my handwriting is distinct from a lower-case w) for beam size for years without any problem. Context makes it clear enough.--Srleffler 01:59, 16 March 2006 (UTC)

First of all, I don't see what the big deal is about using uppercase W versus lowercase w. It's the same letter in the same alphabet. It's not like we are debating whether to use f or ν (that's Greek nu, not Roman vee) to represent frequency, which are two different letters from two different alphabets.

That being said, I would prefer to use the uppercase W instead of the lowercase w for the reason that Srleffler has already attributed to me: namely, lowercase w does appear quite similar to the Greek letter ω, particularly in the font that shows up in LaTeX equations on WP. Now, you have argued that you have no difficulty telling them apart based on context. I have little difficulty figuring it out either, because I have a lot of experience in physics and electrical engineering. I am not concerned, however, for my own sake; and given Srleffler's extensive knowledge of this topic and other optics-related concepts, I am really not concerned about his ability to tell the difference either. I am concerned about people who are learning this material for the first time, who may or may not readily distinguish between the Greek ω and the Roman w.

I encounter undergraduate students almost every day who do not know the difference between these two letters for the simple reason that they have not had much exposure to the Greek alphabet and in particular to angular frequency, which is often represented by ω. Then, after they finally start to get the hang of it, suddenly someone reverses course on them and starts using w in the context of optics, which is of course related to electromagnetic waves, which are of course characterized by angular frequency. And now they see w and think its ω!

Again, it's not a problem for me. I can recognize it quite readily by context and even from the font, because I am aware of this pitfall. But ask a freshman or sophomore who is just starting out in physics or engineering, and you will find that they typically haven't had enough exposure yet to know the difference.

Finally, I do not find it very compelling to argue that we should use w because that is what the image accompanying the article uses. That is clearly putting the cart before the horse. If it makes sense to use W, then we should use W, and we can arrange to have the image revised. If, on the other hand, it makes sense to use w, then we should use w. But it has nothing to do with the fact that the image currently shows it one way or the other.

Now, if someone would like to provide a valid reason why lowercase w makes more sense than uppercase W, I would be more than willing to listen to that argument and give it consideration.

-- Metacomet 05:18, 16 March 2006 (UTC)

When I made the image, I went with w simply because the reference I used (Koechner) uses that notation, as do my old course notes. If there's a consense to change to W I can easily change the image to match, though I have no strong preference either way. --Bob Mellish 05:41, 16 March 2006 (UTC)

Lowercase w seems to be the more common notation--Srleffler 12:26, 16 March 2006 (UTC)

Question for Srleffler (or others): Do you think that using uppercase W will cause any great confusion to the typical WP reader even though the "more common notation" is lowercase w? In other words, do you think that people will have much difficulty realizing that uppercase W represents the same thing that they might have (or might not have) seen in a reference book on this topic? I think you can tell what my opinion is, but please think about it and answer this question honestly. -- Metacomet 14:46, 16 March 2006 (UTC)

BTW, your statement that lowercase is more common may or may not be true. You are correct to note that Saleh and Teich use uppercase W. So from what we know on this page, there are two books that use lowercase, and one book that uses uppercase. Although that is a two-to-one margin, it is not exactly a huge sample size. And if I happen to find just one more reference that uses uppercase W, then the picture changes significantly. -- Metacomet 14:49, 16 March 2006 (UTC)

I, as well, do not care whether it is W or w, but i want it to be consistant between the cross referenced articles that refer to this quantity. In addition to this article that includes at least [Complex_beam_parameter] and [Ray_transfer_matrix_analysis], but there might be others. So if you/we revert it back to W(z), make sure to change these articles as well. --Pmg 06:57, 17 March 2006 (UTC)

[edit] L'Hôpital's rule

Jones, I don't see why you insist on showing the first step in the derivation of the peak intensity. It really doesn't seem all that important. In fact, I think that mentioning that the derivation is done using l'Hôpital's rule is already too much information. Notwithstanding the fact that some other articles show derivations, Wikipedia is not a textbook or an instruction manual. Derivations are really not appropriate content, especially when the derviation is not absolutely central to the subject of the article.--Srleffler 00:43, 23 October 2006 (UTC)