Talk:Helmholtz reciprocity

Why not a merge/redirect?

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

The result of the discussion was no consensus.

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How is this is different from the principle stated (and proved) more precisely in Reciprocity (electromagnetism)?

A lot of the cited sources seem to be describing "ordinary" Lorentz electromagnetic reciprocity (which was generalized from the work of Helmholtz and others), which is also the relevant principle for Kirchhoff's law & the 2nd law of thermodynamics applied to thermal radiation.

Is there any modern source that even uses the term "Helmholtz reciprocity"?

Seems like this should just be a redirect.

— Steven G. Johnson (talk) 20:48, 29 December 2011 (UTC)

The Wikipedia article Reciprocity (electromagnetism) contains the word optic only as follows:

'Lossless magneto-optic materials'

One case in which ε is not a symmetric matrix is for magneto-optic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magneto-optic materials, but restrict ourselves to the situation where material absorption is negligible, then ε and μ are in general 3×3 complex Hermitian matrices. In this case the operator $\nabla \times {\frac {1}{\mu }}\nabla \times -(\omega ^{2}/c^{2})\varepsilon$ is Hermitian under the conjugated inner product $(\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} ^{*}\cdot \mathbf {G} \,dV$, and a variant of the reciprocity theorem still holds:

$-\int _{V}\left[\mathbf {J} _{1}^{*}\cdot \mathbf {E} _{2}+\mathbf {E} _{1}^{*}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}^{*}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1}^{*}\right]\cdot \mathbf {dA}$

where the sign changes come from the $1/i\omega$ in the equation above, which makes the operator ${\hat {O}}$ anti-Hermitian (neglecting surface terms). For the special case of $\mathbf {J} _{1}=\mathbf {J} _{2}$, this gives a re-statement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): the time-average rate of work done by the current (given by the real part of $-\mathbf {J} ^{*}\cdot \mathbf {E}$) is equal to the time-average outward flux of power (the integral of the Poynting vector). By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a Rayleigh-Carson form does not hold without additional assumptions.

The fact that magneto-optic materials break Rayleigh-Carson reciprocity is the key to devices such as Faraday isolators and circulators. A current on one side of a Faraday isolator produces a field on the other side but not vice-versa.

The Wikipedia article on Reciprocity (electromagnetism) contains the word Helmholtz only as follows:

Perhaps the most common and general such theorem is Lorentz reciprocity (and its various special cases such as Rayleigh-Carson reciprocity), named after work by Hendrik Lorentz in 1896 following analogous results regarding sound by Lord Rayleigh and Helmholtz (Potton, 2004). Loosely, it states that the relationship between an oscillating current and the resulting electric field is unchanged if one interchanges the points where the current is placed and where the field is measured. For the specific case of an electrical network, it is sometimes phrased as the statement that voltages and currents at different points in the network can be interchanged. More technically, it follows that the mutual impedance of a first circuit due to a second is the same as the mutual impedance of the second circuit due to the first.

In the early days of Stokes and Helmholtz it was not known that light is an electromagnetic phenomenon. For present-day experts in electromagnetism, the optical implications of electromagetic reciprocity and the meaning of the term Helmholtz reciprocity encountered in older literature on optics and thermal radiation are perhaps trivially obvious, and its relevance to Kirchhoff's law of radiation and the second law of thermodynamics equally obvious. Perhaps some readers of the Wikipedia may not be experts in electromagnetism, and still needing to look up something in the Wikipedia. For such readers, a mere re-direct would likely not be immediately enough what they need to find out what is meant by the Helmholtz reciprocity principle in optics. Perhaps some readers of the Wikipedia article on Kirchhoff's law of thermal radiation may be learning about it for the first time.

On page 21, Chandrasekhar (1950) refers to "Helmholtz's principle of reciprocity", and he discusses it in several places later in his book. In some Wikipedia articles on radiation, Chandrasekhar (1950) is regarded as a reliable source.

According to Hapke (1993) on page 263: "A powerful and useful theorem in reflectance work is the principle of reciprocity, which was first formulated by Helmholtz (Minnaert, 1941)." Hapke is right for the principle for polarized rays, but Stokes had the main idea, though without reference to polarization, a little earlier. Hapke goes on to state the principle for optics, taking a fair-lengthed paragraph to do so. Apparently he does not think the principle is so obvious to his intended readers (advanced undergraduates and beginning graduate students in the physical sciences) as not to need explicit statement.

According to Born and Wolf (seventh edition, 1999) on page 423: "Returning now to the Fresnel–Kirchhoff diffraction formula (17), we note that it is symmetrical with respect to the source and the point of observation. This implies that a point source at P₀ will produce at P the same effect as a point source of equal intensity placed at P will produce at P₀. This result is sometimes known as the reciprocity theorem (or the reversion theorem) of Helmholtz."

In the present Wikipedia article on Helmholtz reciprocity there is a link to Reciprocity (electromagnetic).

It may be useful for the present Wikipedia article on Reciprocity (electromagnetic) to have a historical section indicating the origin of the principle in optics before its electromagnetic character was known.

If the Wikipedia article on Reciprocity (electromagnetic) contained more or less the contents, or an improved version, of this present Wikipedia article on Helmholtz reciprocity in optics at a suitable level of accessibility for the inexpert, then perhaps a re-direct would be good.Chjoaygame (talk) 00:50, 30 December 2011 (UTC)

We should not have a different article on something that refers to the same theorem just because you don't like the other article or just because some authors use a different name for the theorem. The correct thing is to improve the other article as needed.

It is specious to say that the other article does not mention "optics"; optics is electromagnetism (and ray optics is just a limiting approximation of the more general electrodynamics). Nor should our presentation be limited to the level of understanding of optics in 1860!

Moreover, the authors you cite don't even precisely use your term Helmholtz reciprocity: they call it the reciprocity theorem or the principle of reciprocity and attribute its origins to Helmholtz. The other article also attributes the origins to Helmholtz (although the full generalization came later).

Therefore I'm putting a merge tag on this article. — Steven G. Johnson (talk) 04:05, 30 December 2011 (UTC)

My comment was that the word optic appeared in the article Reciprocity (electromagnetism) only in a certain way. I did not write that the article does not mention optics, as you imply that I did. Besides your comment being misleading, your word "specious" implies bad faith on my part, and is therefore contrary to Wikipedia good manners.

I did not propose that our presentation should be limited to the level of understanding of optics in 1860. I was just drawing attention to the relevant range of approaches to optics.

I did not start the article or create its title Helmholtz reciprocity, and I wonder why you seem to feel you should infer that I did.

I have no objection to the article on Reciprocity (electromagnetism), and I wonder why you feel you should infer that I do.

Optics is not the same as electromagnetism. Optics is about the generation and propagation of far-field nearly sinusoidal oscillatory electromagnetic radiation, while electromagnetism includes not only optics but also electrostatics, magnetostatics, classical electric currents, and near- and intermediate-field electromagnetism that does not have to be nearly sinusoidal. Much of optics is dealt with by radiometry, which is not expressed directly in terms of electromagnetism. Much of optics is about quantum effects which cannot be expressed in terms of the classical electromagnetism used in the article on Reciprocity (electromagnetism).

It is clear that the article on Reciprocity (electromagnetism) is written by and perhaps for experts in electromagnetism.

Some Wikipedia readers are not experts in electromagnetism, and it will be far from obvious to them how the reciprocity described in the article on Reciprocity (electromagnetism) relates to the optical phenomena referred to by Stokes, Helmholtz, Kirchhoff, and Planck, that they may interested in.

I know of no obstacle to improving the Reciprocity (electromagnetism) article as needed, as I suggested above should be done in preparation for a re-direct.Chjoaygame (talk) 14:17, 30 December 2011 (UTC)

The question is whether this article should be merged into Reciprocity (electromagnetism). You are (apparently) arguing that we should keep this one on the basis of not liking the level of presentation of the other one, and furthermore continue to imply that it is somehow physically distinct on the basis of it being about "optics" rather than "electromagnetism". Whether you initiated this article is irrelevant; I was responding to your arguments:

• Wikipedia does not have separate articles on the same topic just to present it in different ways. We attempt to have one article on each topic. If the article becomes too long, we split it into a general article plus ones on specific subtopics, not into parallel articles, and especially not into parallel articles that don't make clear the connection to the general topic. If you think the general article on reciprocity is missing something, that is an argument for a merge, not for a separate article.

• Reciprocity is a property of classical optics, which is a subset of classical electromagnetism. Hence reciprocity in optics is not a distinct physical concept, merely a special case of electromagnetic reciprocity. Presenting it as a distinct physical concept, as opposed to a special case of a more general theorem, is a serious mistake in my mind.

Saying that an argument is "specious" means that (to quote a dictionary) it is "superficially plausible, but actually wrong"; it is not an accusation of bad faith.

— Steven G. Johnson (talk) 14:39, 30 December 2011 (UTC)

Perhaps I should start by repeating that I have no objection to the article on Reciprocity (electromagnetism). I note that it does not say much about the use of the general form of Tellegen's theorem to prove reciprocity, inter-reciprocity, and anti-reciprocity theorems (Penfield, P., Jr, Spence, R., Duinker, S. (1970), Tellegen's Theorem and Electrical Networks, Research Monograph No. 58, MIT Press, Cambridge MA, ISBN 9780262160322.)

I did not argue against a re-direct, which I suppose would follow from a merger. I agreed with your statement that "The correct thing is to improve the other article as needed" when I wrote "I know of no obstacle to improving the Reciprocity (electromagnetism) article as needed, as I suggested above should be done in preparation for a re-direct."

I did not argue that reciprocity in optics is or should be presented as a distinct physical concept. I observed that some Wikipedia readers are not experts in electromagnetism.

As expressed in the article on Reciprocity (electromagnetism), electromagnetism is stated in terms of electric and magnetic quantities, while, for purposes like Kirchhoff's law of thermal radiation, and the global illumination algorithms that interested the originator of the Helmholtz reciprocity article, the relevant optics is mostly stated in terms of radiometry. The connection between the two is trivially obvious to authorities such as you, but will not be so obvious to inexpert Wikipedia readers, who do exist, I think. The article on Reciprocity (electromagnetism), in its announced terms of electric and magnetic quantities, is indeed as you say precise; but not in radiometric terms.

I should have made myself clearer. Let me try again as follows.

I think for the benefit of readers inexpert in electromagnetism who look up the Wikipedia in order to find out a little more about the reciprocity that they encountered as attributed to Helmholtz (for example in an article "Helmholtz reciprocity: its validity and application to reflectometry", Clarke, F.J.J. et al., Lighting Research Technology, 17(1):1–12 (1985), or in a textbook such as Applied Optics. A Guide to Optical System Design, Volume 1, Levi, l., Wiley (1968), which writes on page 84: "This implies the Helmholtz reciprocity (or reversion) theorem", it would be good to include in the Reciprocity (electromagnetism) article an expression for optics explicitly in terms of radiometric variables. Indeed, Helmholtz's own expression of it is not too bad. (Re-stating it, Kirchhoff slightly altered Helmholtz's notation, and in the process made a slight mistake that is easily corrected.)Chjoaygame (talk) 19:56, 30 December 2011 (UTC)

Okay, so you agree that it should be merged? (It would have been easier if you'd said this to begin with.) Provided that the other article mentions that some authors use the term Helmholtz reciprocity for the concept?

And, of course, it is perfectly reasonable during the merge to expand the other article to explicitly mention applications to ray optics, and Kirchhoff's law, etcetera. And, as you note, there are other ways the article could be expanded—my argument was never that the other article is perfect, but rather that we should not have two parallel articles on what is essentially the same thing.

(Note, however, that one needs to be somewhat careful in defining reciprocity purely in terms of scalar radiometry quantities like intensity etcetera, because of the polarization/orientation dependence of radiative and scattering processes. For example, the statement here that If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the sensor and light source and the measurement of flux would remain equal seems to require some qualification about the polarization of the light source and the orientations of the sensor and sources in order to be strictly true.)

The main remaining question I have, since I haven't come across this terminology before, is what specifically do people mean when they say "Helmholtz reciprocity" in modern sources like the ones you cite. Do they use it as a synonym for the general Lorentz reciprocity theorem? Or for the Rayleigh-Carson special case? Or for an even more restricted special case, in the ray-optics regime only? Or...? — Steven G. Johnson (talk) 02:02, 31 December 2011 (UTC)

I didn't write the sentence If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the sensor and light source and the measurement of flux would remain equal. I don't try to correct every perception of imperfection in the entries of prior Wikipedia editors. I note that you tell me what I could have said to make things easier.

Clarke and Parry (1985) reproduce and slightly correct Southall's 1924 translation of the third edition of Helmholtz. In the article Helmholtz reciprocity I copy (and slightly correct) Guthrie's 1860 translation of Kirchhoff's slight re-write of Helmholtz as cited by Planck who was writing about Kirchhoff's work. Helmholtz tells about optical polarization in terms of specific intensity (the old term for spectral radiance). Writers who cite Helmholtz, so far as I recall, intend no more than what Helmholtz wrote, in particular not the general Lorentz reciprocity theorem.Chjoaygame (talk) 03:11, 31 December 2011 (UTC)

Potton's 2004 review article on "Reciprocity in optics" (in Reports on Progress in Physics) reviews the work of Helmholtz, Rayleigh, Born and Wolf, Chandresekhar, etcetera, and gives a ray-optical reciprocity theorem (where incident and scattered "rays" are precisely plane waves) as:

The scattering amplitude for a B polarized scattered wave in the direction β arising from an A polarized incident wave in the direction α is equal to the amplitude for an A polarized scattered wave in the direction −α from a B polarized incident wave in the direction −β.

Potton credits this result to de Hoop in 1959, saying that much of the previous work in geometrical optics did not fully take into account the vector nature of the fields. In particular, he seems to say that the derivations of Helmholtz, Rayleigh, and even Born and Wolf were restricted to the scalar (or semivectorial) wave approximation (valid in electromagnetism under very restrictive assumptions). (Of course, the Lorentz reciprocity formulation in terms of currents and fields is fully vectorial, but apparently it took some time for a similar level of generality to percolate into geometrical optics. A bit surprising to me that this took until 1959 since the derivation of the above statement from Lorentz reciprocity seems very straightforward, but on other hand we now have mathematical tools like delta functions that weren't rigorously available until ~1950, and scalar approximations were much more ubiquitous in electromagnetism prior to the last couple decades. Potton points out that Chandrasekhar, in particular, was especially interested in understanding light scattering in planetary atmospheres, where a scalar or semivectorial approximation is valid due to the low index contrast.)

Echoing my concerns with your preference for a radiosity formulation, Potton writes: However, the transverse character of electromagnetic waves adds considerably to the complications of analysis (Perrin 1942). It requires that the exact meaning of exchanging the positions of source and detector is spelt out in a particular experimental configuration.

— Steven G. Johnson (talk) 04:38, 31 December 2011 (UTC)

It is good that we have talked this over.

Contrary to Potton's page 726 perhaps misleading citing of Greffet and Nieto-Vesperinas 1998, Kirchhoff's (Potton's repeated mis-spelling corrected) law has important physics that needs consideration beyond reciprocity alone. Thermodynamic equilibrium is key.

The traditional optical quantity is mainly specific intensity aka spectral radiance, not radiosity.

The quote from page 725 of Potton does not quite tell the story of interest to students of Kirchhoff's law and Planck's law. Rather than concern with possible reciprocity of reflection, r = r′ for Figure 5, they want to know that transmission is reciprocal in the sense that t = t′′ as considered at the bottom of page 734; I do not know the exact reach of Potton's concern that the media should be semi-infinite, but I do not feel too worried about it. Stokes did not consider polarization in his 1849 formulation of the principle for optics. Helmholtz of course does not consider polarization for sound. But Helmholtz didn't wait till 1959 for optical polarization; he did it properly in 1856, and he agrees with Potton's result that Potton says does not come from de Hoop as cited on page 725. Helmholtz notes that magnetism changes things. Chandrasekhar 1950 was perhaps limited in his interests, but still did it properly for optical polarization on page 172 et seq.. I do not see much wrong with simply quoting Helmholtz on optics for this purpose.Chjoaygame (talk) 09:23, 31 December 2011 (UTC)

You can formulate a reciprocity law that includes polarization using a semivectorial approach, and the law may even be correct in general. However, you have not proved it in general unless you use the full vector wave equations. It seems unlikely to me that Helmholtz did that before Maxwell wrote down Maxwell's equations. — Steven G. Johnson (talk) 20:12, 31 December 2011 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

proposed reasoning

What do you think of the following reasoning?

Optics is about far-field activity propagating far from its source, relative to the wavelength. The time course is nearly sinusoidal, and this holds also for unpolarized sound signals. The result for light is that the electric and magnetic vectors are as it were exchanging energy in a balanced way, so that for a light ray, knowledge of one is very informative, nearly completely informative, about the other. Light polarization is describable in terms of helicity. Just two components are needed, the left helicity and the right helicity, with their relative phase. These two components carry the same information as the two fields, electric and magnetic, under the constrained conditions of a light ray. This extends to the picture of light in terms of two directions of plane polarization. Also the polarization of a polarized ray is preserved under suitable conditions, and the two polarizations do not somehow cancel or de-phase or otherwise interfere with each other. Thus a light ray can be described as consisting of two radiometrically measured components, one for each of a chosen pair of polarization directions; this much was I think assumed by Helmholtz on empirical grounds; perhaps he did not think in terms of electromagnetism, but just in terms of sinusoidal propagation of two preserved components of a ray.

How does this relate to reciprocity? The usual form of reciprocity would be that the magnetic outcome is reciprocal with the electric source. So to get reciprocity of transmission, one would say that the magnetic signal should be put into the outcome channel and the original electric source would be recovered at the input. But the close linkage of electric and magnetic signals for light, as noted above, means that one might just as well put the original electric signal back into the outcome channel, and expect to recover the original magnetic outcome signal from the input channel. Thus a polarized component light ray goes equally in either sense along a given path. That is all that Helmholtz is claiming, I think.Chjoaygame (talk) 04:08, 1 January 2012 (UTC)

Great article, anyway. 84.227.255.16 (talk) 08:25, 16 September 2014 (UTC)