# Talk:N-slit interferometric equation

The intro is quite clear: this is a generalization of Feynman's double-slit interference which is done using Dirac's notation. The basics are explained in the Feynman Lectures on Physics or Dirac's book. The sign

is unnecessary. (Note by Nanite: the preceding comment was added by 128.151.39.151)

I disagree with the preceeding comment. The article is about the N-slit interferometrc equation; ths can be modelled by classical wave equations as well as Dirac equations so why is the introduction mainly concerned with Dirac and Feynman.
And BTW, what is the different between N-slit interference and diffraction by a grating? Epzcaw (talk) 17:05, 4 October 2013 (UTC)

## Merge with Fraunhofer diffraction

It's not entirely clear what distinguishes the topic from Fraunhofer diffraction (mathematics), as the mathematics has a very close resemblance. It is unusual that both the formulations and applications parts cite mainly articles by F. J. Duarte, as it makes this article seem either extremely specialized (that a diversity of sources cannot be found), or biased towards his viewpoint.

If this article stays, the lead section should be rewritten so it does not read like a journal article abstract. The first sentence should be "The N-slit interferometric equation is used to calculate _____", at a level accessible for anyone who might be interested in this topic.
--Nanite (talk) 10:27, 24 June 2013 (UTC)

There is another article entitled N-slit interferometer which is covering the same material in more detail. Shouldn't these two article be merged?Epzcaw (talk) 16:43, 4 October 2013 (UTC)

## Merge with Fraunhofer?

Not a good idea. It would result in a messy and confusing page. There are several differences between the Fraunhofer approach and the Dirac approach. As explained in books like Fowles's the Fraunhofer method is a purely classical formalism that mainly applies to plane wave illumination. The Dirac approach applies to either plane illumination or highly diffractive patterns as explained in the references of N-slit interferometric equation. The Dirac method also applies to single-photon propagation (Fraunhofer's does not) and indistinguishable photons. The later may be an advantage or disadvantage depending on the application.

As the main contributor to N-slit interferometric equation I think that deletion of this article would be preferable to a merger. Corrigendas (talk) 19:49, 27 June 2013 (UTC)

I have just seen these articles. As the main contributor to the Fraunhofer (mathematics) article, I also think that that the articles should not be merged, as they are looking at different things. However, they could be better cross-referenced, and I will have a go at doing this in my sandbox.
Re your comment above - if there is something which is explained in Fowle's book but not here, wouldn't it be a good idea to include it here?
Also, I don't understand your comment that Fraunhofer's method does not apply to a single photon - all optical interference and diffraction results apply only to a single photon , just as with Young's slits, which has been shown to work with single photons - see Double-slit experiment Epzcaw (talk) 16:54, 4 October 2013 (UTC)
I agree that these articles should have better cross references and that the differences between the two methods should be nicely explained. However, this particular article is about a theoretical tool, an equation, and not about an optical element. A diffraction grating is not an equation, a diffraction grating is an optical element. Advocating the merger between an article devoted to an equation and an article on an optical element... should surely greatly confuse the readers.Corrigendas (talk) 02:18, 6 October 2013 (UTC)
An N-slit interferometer is an optical element which is (as far as I can see) the same thing as a diffraction grating, so the equation(s) describing how light is affected by it must be identical to the equation(s) governing how light is affected by a diffraction grating. If not, why not? If so, this should be made clear, an reference made to the other articles discussing diffraction gratings.
The article on its own provides very little useful information. Most of the terms in the equation are undefined. It would be a major improvement if the complete derivation were given, rather than "after some algebra". Without these, it is not possible to see what the difference is between this and the classical diffraction equation of Kirchoff-Fresnel (rather than Fraunhofer which is a far-field approximation).
Similarly, the diagram is mainly unexplained. What are TBE, MPBE, D?
Finally, if this is an article about an equation why does it have an application section - this overlaps the N-slit interferometer article which you don't think should be merged with this one. Many articles include theory and applications so I'm not sure what your objection since it already occurs here!!! Epzcaw (talk) 09:32, 8 October 2013 (UTC)
The additional equations are an improvement. However, shouldn't you have

$\Psi(r_j) = \Psi(r_{x,j}) \Psi(r_{j,s}) e^{\Omega_j}$ where

$\Omega_j = \theta_j+\phi_j$
Epzcaw (talk) 14:13, 12 October 2013 (UTC)

The math should be clear now.Corrigendas (talk) 19:57, 12 October 2013 (UTC)
Now that I can see how the derivation works, I cannot see how the mathematical formalism differs in any way from classical wave optics.
The amplitude of the light field at the jth slit from an extended source can be described by Kirchhoff's diffraction formula
 $U(P_j) = - \frac {1}{4 \pi} \int_{A_1} \frac {e^{iks}}{s} \left[ ik U_0(r) \cos(n,s) + \frac {\partial U_0(r)}{\partial n} \right] dS$
This is a complex number which can be represented by
$\Psi(r_{j,s}) e^{-i \theta _j}$
The amplitude of the light field arriving at x from the slit can again be described by using the Kirchoff diffraction formula - each component of the light wave at the slit is diffracted in the way described by that equation, and after some messy sums, we get an expression for the amplitude of the light arriving at x via the jth slit as
$\Psi(r_j) = \Psi(r_{x,j}) e^{\theta_j }\Psi(r_{j,s}) e^{\phi_j}$
The overall amplitude light field is then given by the sum of the contributions from the individual slits, just as in N-slit interference equation, and the intensity I(x) is given by the amplitude multiplied by its complex conjugate, giving
$I(x)= \sum_{j=1}^\N\,\Psi(r_j)^2\ +2 \sum_{j=1}^\N\,\Psi(r_j)\bigg(\sum_{m=j+1}^\N\,\Psi(r_m)\cos(\Omega_m-\Omega_j)\bigg)$
The difficult part is then solving this equation, and whether you arrived at it using classical wave theory or Dirac's notation is neither here nor there - the solution will be the same.
Fresnel and Fraunhofer are approximations used to give solutions for the (fairly) near field and the far field. Analytic solutions are not available for the very near field or for complex arrangements but a variety of numerical modelling techniques are available which use classical wave theory (e.g. Boundary element, Transmission line matrix) which provide solutions for almost any configuration.
Your comparison with classical methods mentions only Fraunhofer and Fresnel, but as the equation above indicates, the (classical) Kirchoff equation can be applied to any configuration - the tricky bit is solving it. Epzcaw (talk) 11:45, 13 October 2013 (UTC)
One of the beauties of the Dirac approach is its mathematical simplicity. This easily allows rather complex calculations as the generation of interference patterns produced by a cascade of N-slit arrays (see ref. 5) and the illumination of either a wide single slit, or an array of N-slits, by highly diffractive patterns.
In his comparison between the Dirac and Fourier methods Taylor (1995, Ref. 8) wrote: "both models are very accurate in the Fresnel and Fraunhofer regime. At regions very near the diffraction screen, the classical model breaks down. However, the Dirac model remains accurate... The main difference of the two approaches, for applications in the macroscopic domain, appears to be a philosophical one." Corrigendas (talk) 17:56, 13 October 2013 (UTC)
The equation is indeed simple, but as it stands, provides no indication of how you do the "rather complex calculations" to solve real problems. I can find out how to try to solve a wave propagation problem using Fraunhofer, Fresnel, Kirchoff, Boundary Element, Transmission Line Matrix modelling etc from the Wikipedia articles on these topics, but I have no idea how the very simplistic equation given here could model anything. I believe you (or someone else) should give the derivation of 2-slit, or 3-slit or even N-slit interference which you say is available in ref 5. We need to know how we derive expressions for $\Psi, \phi, \theta$ in order to calculate the overall interference pattern amplitude, intensity, probability amplitude, call it what you will.
As I said before, Wikipedia articles should be self-contained, and not simply a list of references to external work. Epzcaw (talk) 14:22, 14 October 2013 (UTC)
The Fraunhofer method is known in the literature since the 1800s and it is treated in many books, so it is only natural to expect to see it well taught. Certainly, a self-contained article should be the aim for this article, however, that takes time.Corrigendas (talk) 02:05, 15 October 2013 (UTC)
I would not want to see this article merged with either Fraunhofer article because in my opinion it does not provide any useful information. The DD equation does not give any help in actually analysing optical systems, and the examples given could readily be solved using classical wave equations. I have looked at one of the papers referred to which is available online, and the analysis therein uses exactly the same geometrical analysis as used in a classical wave model. The comparison between the DD equation and classical wave equations is meaningless since it neglects the large part of classical wave theory. The implication is that only far-field and fairly near field systems can be modelled using classical wave theory but this is patently incorrect - see next section. Epzcaw (talk) 16:57, 20 October 2013 (UTC)
Good to see that there is no support for the merger. As far as this article not providing "useful information", that is rather subjective. The fact of the matter is that generalized interferometric calculations, near or far field, can now be easily performed without having to use the old classical formalisms. Corrigendas (talk) 19:49, 20 October 2013 (UTC)
Until some examples are provided, there is no evidence that interferometric calculations can be performed any more easily with this formalism than the classical one.. When I have time, I will reproduce the stuff in the paper by Duarte which I found online, to show that the calculations are identical to the classical calculations. It would be good to see a simple 'very near field ' calculation, but I doubt if such a thing exists as the field, by its very nature is extremely complicated. How about a derivation of the wave field at a distance of 1/100th of a wavelength from your N-slit device? Epzcaw (talk) 19:59, 21 October 2013 (UTC)
A comparison, between the DD approach and the K approach, may be publishable.Corrigendas (talk) 01:09, 22 October 2013 (UTC)

## N-slit interferometric equation = interferometer?

It has been suggested that the N-slit interferometric equation article be merged with the article on the N-slit interferometer. This article is about a generalized N-slit interferometric equation that can be used, among various applications, to describe the N-slit interferometer. The merger suggestion, although well intentioned, confuses the divide between technological instruments and the theory developed to describe them.Corrigendas (talk) 02:50, 6 October 2013 (UTC)

## Comparison with classical methods - criticism

This section is not accurate as the comparison is made a very set of classical wave theory models.

1) It assumes that the only classical models are Fraunhofer and Fresnel. The basic classical wave propagation model is Kirchhoff's diffraction formula of which the Huygens–Fresnel principle is an approximation. This can be applied to any wave propagation system - though is not always solvable analytically.

2) It ignores the variety of numerical modelling systems based on classical wave theory, e.g, Boundary Element, Transmission Line Matrix which enable almost any configuration to be modelled if sufficient computing power is available - very near field systems can be readily analysed and they work for any form of illumination.

3) it claims that the DD equation is more compact, but this is only the case because it omits any useful information about the system being modelled.

4) As the article gives no explanation about how the DD can be solved, it is not clear how the DD equation is used to model any wave propagation system, let alone the complex systems mentioned above Epzcaw (talk) 09:32, 16 October 2013 (UTC)

The comparison only refers to Fraunhofer and Fresnel. As far as I know no one has made a comparison with the Kirchhoff's approach. Also, this comparison was motivated by comments on this page and is not important to the article. The comparison can be either improved, reduced in scope, or even bypassed altogether.Corrigendas (talk) 01:35, 18 October 2013 (UTC)