Talk:Fraunhofer diffraction equation

Direction coordinates

This: "where l,m are the direction cosines of the point x,y with respect to the origin" was clear as mud to me. I've never heard the term 'direction cosines' before, linked wiki article on direction cosines did not help. For one thing, which origin is it in respect to - the primed or unprimed? Wouldn't it be easier to just say "where l = x/z and m = y/z"? (I don't want to make the edit yet in case I still misunderstand). Argentum2f (talk) 18:09, 20 March 2019 (UTC)

I agree, direction cosine here makes no sense. It seems more like a "direction tangent" to me, but perhaps I also misunderstand. Direction cosine would seem to be something more like l=x/r, i.e. roughly 1-x²/2 I'll ponder over it a little more then perhaps make the edit. Imyxh (talk) 21:14, 27 June 2022 (UTC)

I agree, too. I see a second form for the equations was added in terms of $x,y$ instead of $l,m$, which I find to be more clear.

Would anyone be opposed to simply deleting the first equations and any use of the variables $l,m$. Whether or not they really are 'direction cosines', they seem unnecessary. Jackmjackm (talk) 18:52, 18 November 2023 (UTC)

First Picture is Wrong

Both vectors r and r' should originate at the origin of the x',y' plane and the vector r' obviously cannot extend beyond the aperture, shown in that figure as a circle. In the past, there was simple derivation of the far-field equations based on the simple idea of a superposition of expanding plane waves, originating from the aperture in the x',y' plane. For unknown reasons, this derivation has since been deleted. 24.85.247.169 (talk) 08:01, 25 January 2012 (UTC) — Preceding unsigned comment added by 157.127.124.15 (talk)

the only line I need on this whole page of advanced mathematics is If the aperture is illuminated by a mono-chromatic plane wave incident in a direction {{math|l0,m0,, n0) and I have no idea what it means. I will research it and get back to you, wikipedia. Also, shouldn't this be in the "Fraunhofer diffraction" page anyway...? Schmave123 (talk) 15:17, 28 September 2011 (UTC)

I have corrected the error which you have pointed out - two"}" were missing. The parameters refer to the direction cosines of the incident wave.

The reason for keeping this page separate from the Fraunhofer diffraction page is that many wiki readers are put off by detailed mathematics, but nonetheless would like to have some understanding of the topic. On the other hand, some readers want to understand the mathematical derivations, and may even want to apply them. This is the reason for providing lots of simple examples of the application of the Fraunhofer diffraction equation.Epzcaw (talk) 11:54, 9 October 2011 (UTC)

strange typos

Probably in hundreds of Wikipedia articles, I've seen

p312

and the like, when what was meant was

p 312

i.e. page 312. As far as I know, this never happens except in Wikipedia articles, and there, it's commonplace. I just fixed a large number of those in this article.

Is there some manual or something saying this non-standard usage should be followed? If not, why does this happen? Michael Hardy (talk) 04:22, 24 January 2012 (UTC)

This is what Wikipedia says:

"On all of these points, Wikipedia does not have a single house style. Editors may choose any option they want; one article need not match what is done in other articles. However, citations within a given article should follow a consistent style." Epzcaw (talk) 15:35, 28 January 2012 (UTC)

Large-angle behaviour

Hi all, I've noticed in these pages that at several points the expression $x/z$ is converted into $\sin \theta$. This is a bit misleading for the following reasons: At small angle we should just say $x/z\approx \theta$, which makes things simpler. At large angles, we need to use the proper expression $x/z\equiv \tan \theta$. Using the plain $\theta$ approximation is actually MORE accurate than sine, since sine bends in totally the wrong way at large angles! — Preceding unsigned comment added by 24.85.247.169 (talk) 08:01, 25 January 2012 (UTC)

Suggested integration of N-slit diffraction article

I have just seen this.

I agree up to a point. Fraunhofer diffraction by an N-slit grating is already given here, but maybe needs some corss-referencing. Will look at it, and put it in my sandbox.

However, the other article seems to relate mainly relate to Quantum mechanics so maybe a modified version needs to stay.Epzcaw (talk) 11:50, 4 October 2013 (UTC)

There is a detailed discussion about this on the N-slit_interferometric_equation article and there appears to be a consensus that two should no be merged. In my opinion, the other article contains no real information, just refers to some papers.

So I propose to remove the suggestion that the N-slit diffraction article is merged with this one, as it has not had any support. Please comment in the next week if you disagree. Epzcaw (talk) 16:45, 4 April 2014 (UTC)

Illustrate the Fraunhofer diffraction of light due to single slit and find the intensity with necessary equations

Ghyh 175.101.68.25 (talk) 17:02, 26 November 2022 (UTC)

Proportionalities instead of equalities

Most of the key formulas are given in terms of proportionalities. Is there a good reason to omit the exact equations?

For example, according to

https://phys.libretexts.org/Bookshelves/Optics/BSc_Optics_(Konijnenberg_Adam_and_Urbach)/06%3A_Scalar_diffraction_optics/6.07%3A_Fresnel_and_Fraunhofer_Approximations

the first equation appears to have a prefactor of ${\frac {e^{ikz}e^{ik(x^{2}+y^{2})/2z}}{i\lambda z}}$. Jackmjackm (talk) 18:46, 18 November 2023 (UTC)

Yes, this would be the prefactor in the Fresnel approximation regime. The reason for the use of proportionalities is because there is the large z approximation (a piece of the definition of the Fresnel approximation regime) and so use of a strong equality would be misleading. In addition when U is a scalar map from R³ to R, this is another piece of the approximation regime, namely, taking the electric field to be approximated by a scalar function. In general, the electric field is properly a vector field (or one form when defined through the appropriate gradient accordingly), and as such would have 3 components.

Another approximation being made is that an image (the resultant of propagating light) is being characterized by only the electric field, while one knows more generally that light is in fact simultaneously both necessarily an oscillating electric field and an oscillating magnetic field. Yet another approximation is ignoring the polarization/spin of light.

Nonetheless when all the right conditions are met, the Fresnel approximation regime can, to reasonable accuracy, predict the resultant image of light diffracting through an (idealized planar) aperture. This points out another approximation, which is that the target is faithfully characterized by its 2d cross section or that it is idealized as planar. MMmpds (talk) 15:24, 19 November 2023 (UTC)

Are you sure you aren't conflating proportionality and approximation?

Proportionalities, denoted by $\propto$, suggest the expression is missing a constant prefactor. And by constant, I mean it doesn't depend on any variables we may care about in a given context. The missing prefactor can be small or large.

Approximations, denoted by $\approx$, suggest the omitted prefactor is close to 1, so it can be neglected with a tolerable amount of error.

Also, are you sure that representing a vector function as a scalar is an ``approximation?" I interpret this scalar function as a single component of a vector field. It's not an approximation; I'd call it a simplification.

According to the website I linked, the factor I provided is the prefactor for both the Fresnel and Fraunhofer regime. The difference between the two is not in the prefactor, but rather the expression within the integrals. Jackmjackm (talk) 16:49, 20 November 2023 (UTC)

So simply put, no, there is no conflation of binary relations. The issue is that the phrase, "... doesn't depend on any variables we may care about in a given context... " has its context defined by the approximation regime. The spatial component $z$ is treated as a large dominant variable, which after approximation concludes with it being a fixed free index. While the spatial variables $x$, and $y$ are constants because they remain fixed free indices throughout.

The squiggly equal sign as a symbol for approximation does not always stand for a missing prefactor close to 1. A common application that does not follow this pattern is the Taylor approximation of an analytic function, which differs by an additive (not multiplicative) term of higher order than the truncation being made in the Taylor approximation.

For more exposition on scalar approximations to the electromagnetic field, see John David Jackson (physicist)'s book "Classical Electrodynamics", 3rd edition (for example). In that edition, chapter 10 is entitled 'Scattering and Diffraction' and contains subsequent sections 10.5 'Scalar Diffraction Theory', and 10.7 'Vectorial Diffration Theory'. Therein, there is extensive discussion about the differences between various scalar approximations, and shows how and under which conditions the Kirchhoff scalar integral can be relied upon. Jackson does this by first constructing the vector analogue of the Kirchhoff integral in 10.6, where it is made clear what a scalar approximation actually means.

So as not to spam the Talk page, if there are further questions or more references sought, feel free to send an email. MMmpds (talk) 19:56, 20 November 2023 (UTC)

Ah, I think we are not completely on the same page. But for the sake of brevity, would you be opposed to me changing the equation from

$${\begin{aligned}U(x,y,z)&\propto \iint _{\text{Aperture}}\,A(x',y')e^{-i{\frac {2\pi }{\lambda z}}(x'x+y'y)}\,dx'\,dy'\end{aligned}}$$

to

$${\begin{aligned}U(x,y,z)&\approx {\frac {e^{ikz}e^{ik(x^{2}+y^{2})/2z}}{i\lambda z}}\iint _{\text{Aperture}}\,A(x',y')e^{-i{\frac {2\pi }{\lambda z}}(x'x+y'y)}\,dx'\,dy'\end{aligned}}$$

It appears to be accurate (I just proved it to myself for a simple Gaussian Beam propagating from its focus to the far field, at least). I find this equation more useful than a proportionality. Jackmjackm (talk) 21:05, 20 November 2023 (UTC)

Of course, one does not need to begin with consensus to make changes, as a general rule regarding Wikipedia. The only comment now is the inconsistency in usage from proportionality to approximation. It is probably the case that this article is best served by a complete re-working starting from the very first sentence. It is unfortunate that some sources take the quadratic exponent prefactor to be 1 owing to the largeness of $z$, while others keep it as is proposed.

The Fraunhofer approximation is a scalar far field approximation and so I happen to feel that the prefactor is best served by simply writing an outgoing spherical wave. This, however, changes the denominator to the 3-Euclidean norm of $x$ (and possible a $2\pi$ depending on the choice of Fourier kernel), while the numerator becomes $(k\cdot {\hat {n}})$. There just doesn't seem to be a single right way of writing things because it depends heavily upon where one wishes to go.

Happy editing! MMmpds (talk) 17:49, 21 November 2023 (UTC)