# Eigenfunctions of the vector Helmholtz equation

• Engineering
Hi everyone,
I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called vector Helmholtz equation and its Neumann and Dirichlet problems.

I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics, vol.2, but that treatment I think is really general (and also quite complicated, for me, to follow) and it separates the solution into three eigenvector groups L, N and M and the solutions will be a linear combinations of the three. For sure I wasn't able to fully understand this treatment, but I'm interested in this kind of theory applied to the electromagnetic context, in particular for resonant cavities.

In my (italian) waveguides book, I can find in an appendix where the solutions of that kind of problem can be divided in three kind: solenoidal, irrotational and harmonic (and this last one is generated when the surface is not simply-connected). Can anyone suggest me a reference for this kind of treatment, oriented to the electromagnetism? The next step is to express these solutions into TE, TM and TEM solutions: how can you connect the solenoidal/irrotational/harmonic siolutions with the TE/TM/TEM? It's just something like solenoidal -> TE, irrotational ->TM, harmonic ->TEM? I'm not sure about this.

Thank you!

vanhees71
Gold Member
2021 Award
As usual Jackson is a good choice when it comes to electromagnetism. He has an entire chapter on the multipole expansion in terms of spherical harmonics. Another nice source about the vector spherical harmonics is

R. Barrera, G. Estevez, J. Giraldo, Vector spherical harmonics and their application to magnetostatics, Eur. J. Phys. 6, 287 (1985).
http://stacks.iop.org/0143-0807/6/i=4/a=014

Hi Vanhees and thank you for your answer. In my free time I still tried to check your suggestions, but I believe they are too complicated for me (I'm an engineer). An engineer-oriented book would be better.

Another argument that I'm looking for and is related to this one is the mathematical justification of the modal expansion of the waveguide field in the Marcuvitz notation. In his books there is almost no justification and the results are given as they are.

If anybody could help I would really appreciate.
Thank you

jasonRF
Gold Member
A mathematical justification of the modal expansion requires that you prove that the modes form a complete set. This is not easy to do. I am familiar with quite a number of graduate level engineering electromagnetics texts that use modal expansions, but do not recall any of them proving completeness. I doubt you will find such a proof in any book that is and simpler than Jackson (who does not prove it) or Morse and Feshbach (which I have only flipped through on a few occasions, so I do not know if those books include such a proof).

Jason

jasonRF
Gold Member
Hi everyone,
I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called vector Helmholtz equation and its Neumann and Dirichlet problems.

I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics, vol.2, but that treatment I think is really general (and also quite complicated, for me, to follow) and it separates the solution into three eigenvector groups L, N and M and the solutions will be a linear combinations of the three. For sure I wasn't able to fully understand this treatment, but I'm interested in this kind of theory applied to the electromagnetic context, in particular for resonant cavities.

In my (italian) waveguides book, I can find in an appendix where the solutions of that kind of problem can be divided in three kind: solenoidal, irrotational and harmonic (and this last one is generated when the surface is not simply-connected). Can anyone suggest me a reference for this kind of treatment, oriented to the electromagnetism? The next step is to express these solutions into TE, TM and TEM solutions: how can you connect the solenoidal/irrotational/harmonic siolutions with the TE/TM/TEM? It's just something like solenoidal -> TE, irrotational ->TM, harmonic ->TEM? I'm not sure about this.

Thank you!
The simplest treatment is probably in "foundations for microwave engineering" by Collin. Both the first and second editions have a chapter on cavities that includes the modal expansion, and divides the modes up into TE and TM (no TEM in the cavity!). It is a little easier to read than Jackson. Pozar might cover this in his "microwave engineering" text, but my copy is at work so I'm not positive about that. At the level of Jackson, I recommend "field theory of guided waves" by Collin, "electromagnetic fields" by Van Bladel, and "time harmonic electromagnetic fields" by Harrington. They are all written for engineers so they may be easier for you to read than Jackson was, even though they are at a similar level. In any case, don't buy any of them without looking at them first - try to find them in your library.

Jason

Thank you Jason. I'm actually pretty familiar with the books that you cited (some of them are in my office too ;) ), but, being a microelectronic engineer and not a microwave engineer, I don't have a profound knowledge of all the math (that is generally a lot!) behind the books' discussions. Also, it's almost 10 years that I don't take a course in electromagnetism, but in the free time I like to read it and learn. However, I have my problems since I try to understand the reasons of hypothesis made in some treatments, while I feel more "safe" when I know that there is a math theory behind every assumption. Functional Analysis in the context of electromagnetism is probably where I should look for math justifications but it's really hard for me to find books that treat it applied in the field (many books treat the theory for scalar functions but very few also for vector functions, and usually in a manner that is not useful or easily understandable for me).

I'll try to explain what I like, what I don't like and what I don't understand in the books that you cited and others. By the way, I'm moving a step back: now I'm taking a look to the modal expansion inside waveguides instead of cavity, that is I believe easier than the modal expansion inside cavities, but where I encounter the same difficulties.

1) I don't like the fact that, in a typical treatment of the modal expansion inside cylindrical wavegudies, the author makes assumptions that are not particulary intuitive to me and, moreover, they are not mathematically justified. As an example, Collin, Ghose, Pozar, Kurokawa and probably others, before starting the treatise, they all make the assumption that the electric/magnetic field is varying exponentially with the cylindrical coordinate z (e.g. pag.97 Collin Foundations, pag. 105 Pozar...). I'm not able to grasp the insight in this hypothesis, for me it is better if I see a differential equation and I know that its solution will be exponential. I know that there is also the time exponential variation, but with this I'm more used from Fourier study. I know that conceptually is the same, but a Fourier expansion in space is less intuitive for me.

2) Another thing that I don't like is the fact that the author already separates solutions into TE, TM and TEM and then treat them separetely: in this way he can introduce a lot of simplifications at the very beginning, but I'm not able to forget to ask myself "yes, but the initial hypothesis of separation into TE, TM and TEM where comes from, unless I already know how the treatise is gonna end?". It seems to me that Collin use this approach.

3) I'll try to explain what instead I DO like: the approach of Marcuvitz. To me it seems that starting from general Maxwell's equations he derive its theory only with the use of mathematical theory and only during the process he is able to distinguish the existence of different modes. Unfortunately, in its Waveguide Handbook the theory is not very well explained, and I tried to reconstruct it looking on different books. A smaller treatment is made by Harrington in the chapter about Microwave Networks.

What it seems to be very much like the Marcuvitz treatment is what I found inside the book by Milton-Schwinger

In chapter 6 there is a much more explained treatise of the Marcuvitz expansion and I really like this type of discussion, since from general theory he gets the usual results. Anyway I have some doubts that I wanted to ask you about these few pages You should be able to see the first pages of the chapter excluding pag 107-8 with google books.
- equations 6.8 and 6.9 seem supported by math theory (transverse field decomposition, even if I remebered only a 3D field decomposition theorem), so I like it
- I absolutely don't understand the observation below eq. 6.9: can you help me? For sure it's related to my holes in functional analysis. Even a reference would be useful.
- on pag. 106 there is the big hole for me: the use of the eigenfunctions of the transverse laplace operator. Can you suggest me a reference where it is demonstrated that a function V(x,y,z) can be decomposed with two function phi(x,y)V'a(z) as an eigenfunction of the tranverse laplace operator and where the hypothesis 6.13 comes from (since I was expecting also the z coordinate in that equation)? This would be very helpful for me.

I believe I wrote everything that was in my mind, I'm sorry for the long post.
I really appreciate if you'll be able to help me.
Thank you in any case.

jasonRF
Gold Member
I am not able to see page 106 of the reference so can only help you a little.

1) the reference you link assumes that the guiding structure is uniform in z, so the transverse and parallel parts of the Helmholz equation are separable. This means that the equation describing the z dependence is the same that Collin (and others) assume (which is also the same operator as the temporal piece that you are comfortable with), so the eigenfunctions of this operator in z are in general complex exponentials. Your linked reference doesn't use this fact until page 110, but it doesn't really make the treatment any more general than if that fact is used earlier in the derivation. Think of the z-invariant structure as begin the spatial analog of a time-invariant system; a complex exponential is the eigenfunction for each, and Fourier analysis lets you construct anything you want from a set of these eigenfunctions.

2) This is a matter of taste; in my view there is no right or wrong here. You are pursuing an approach that doesn't decompose the solutions on these physical grounds, which is reasonable on your part. I personally think that Collin's foundations book does a good job, although he doesn't prove that for many structures of interest not all components of the fields are required to satisfy the boundary conditions (Ez=0 and/or Hz=0 is allowed). He simply states that fact, and then uses it to justify completely separating the TE and TM modes. But even if it weren't true, the linearity of the differential equations means that a solution with Ez=0 can be added to a solution with Hz=0 prior to applying the boundary conditions. A general field that isn't a single mode would in general need both anyway. This is essentially what Milton-Schwinger are doing. It is interesting to me to look at it from a different approach, so I'm glad you bring it up.

3). The observations below equation 6.9 has nothing to do with functional analysis (which may be required to prove completeness ...). Anyway, as far as I can tell the authors are alluding to equations 6.7. I would recommend trying to separate out the perpendicular and parallel parts of the fields from those equations and see if their comments follow. I have a feeling they are making this more complicated than it needs to be (perhaps by not assuming $e^{\pm i \beta z}$ at the outset).

Regarding your comment about the separation on page 106 which I cannot see - if you write out using latex I might understand what your notation means (what does the V' in phi(x,y)V'a(z) mean? What is the equation it is supposed to solve?). But I have a hunch this simply has to do with the assumption that the structure is uniform in z, so the solutions naturally separate as one function of x and y, times a function of only z.

Sorry I cannot be much help here!

Jason

vanhees71
Gold Member
2021 Award
Well, there's no other way to understand this stuff than to learn the math. It's not that difficult. You need Helmholtz's fundamental theorem and generalized Fourier transformations (i.e., the expansion of the fields in terms of the appropriate mode functions of the wave guide or cavity).

Thanks everyone.
Vanhees71, I'm familiar with Fourier transform when talking about time functions of time only and when talking about generalized functions in Hilbert spaces let's say that I know the theory, but the big jump is moving from theory to practice, especially when many coordinates are involved (more than 2). A book that could be useful in this sense could be Operator Theory for Electromagnetics by Hanson, but I still have to check it properly.

For Jason:
1) the reference you link assumes that the guiding structure is uniform in z, so the transverse and parallel parts of the Helmholz equation are separable.
That's the "so" part that is not immediate to me. I mean, I don't have the full background to notice particular structure of the coordinate system/boundaries and then apply these info to solve the partial diff equation.
Think of the z-invariant structure as begin the spatial analog of a time-invariant system; a complex exponential is the eigenfunction for each, and Fourier analysis lets you construct anything you want from a set of these eigenfunctions.
This instead is a very interesting point of view! Thank you, I'll study in deep.

3) I'll try to rephrase my doubts about what the authors of the book are saying in that observation: looking at equation 6.7 $$\frac{\partial}{\partial t} \mathbf{H} = ik \eta \left( 1+ \frac{1}{k^2} \nabla \nabla \right) \cdot \mathbf{e} \times \mathbf{E}$$ where $\mathbf{e} \times \mathbf{E} = \mathbf{E}_{\perp}$ they say that can be rewritten using equation 6.8, but honestly I still don't understand the development of this representation that is described in the observation. E.g. where the condition 6.10 $\nabla_{\perp} \nabla \mathbf{A}_{\perp} = \gamma \mathbf{A}_{\perp}$ is taken from? Is just a rephrasing of $\left( 1+ \frac{1}{k^2} \nabla \nabla + \gamma \right) \mathbf{A}_{\perp} = 0$, that should be the definition of the $1+ \frac{1}{k^2}\nabla \nabla$ eigenproblem? Am I making a lot of confusion here or I'm somehow right? Maybe it's just a matter of making some vector computation and rewriting using vector identities the expression $\left( 1+ \frac{1}{k^2} \nabla \nabla \right) \cdot \mathbf{e} \times \mathbf{E}$, making some observations? This puzzles me.

This is the screen of pag.106: http://imgur.com/a/Bpw9l
Notice that it is with equation 6.12 that becomes evident the separation of TE and TM modes (and not at the very beginning of the treatise). I very much prefer this approach: of course, as you correctly say, it's a matter of taste.
I'll try to interprete what happens from line "the scalar quantities involved" with my own words. Basically, since the operator $\nabla^2_{\perp}$ is involved in the equation, it is possible to substitute to each, say, $\nabla^2_{\perp} V'$ appearing in equations 6.11 its definition of the eigenvalue problem $$\left(\nabla^2_{\perp} + \gamma^2_{a} \right) V'(x,y,z) = 0,$$ and the expansion in eigenfunctions $$V'= \sum_{a} \varphi_{a}(x,y) V'_{a}(z),$$ that of course I take as it is since I don't know where to find the proof of this particular decomposition. The condition 6.13 $\left(\nabla^2_{\perp} + \gamma^2_{a} \right) \varphi_{a}(x,y) = 0$ comes from $\left(\nabla^2_{\perp} + \gamma^2_{a} \right) V'(x,y,z) = 0,$ supposing that $V'$ is expanded as shown above, am I correct? Moreover, this separation of variables (z alone while x,y together instead of x,z together and y alone, for example) is intrinsic in the coordinate system and it is independent from boundary conditions? Because when the authors say "The final step in the reduction to one-dimensional equations consists in representing the x, y dependence of these functions by an expansion in the complete set of functions forming the eigenfunctions of $\nabla^2_{\perp}$", besides not saying anything of the role of the z coordinate in the expansion, they don't also considerate boundary conditions.

GENERAL FUNCTIONAL ANALYSIS DOUBT: when I know that an operator has an eigenvalue problem definition and can be written in a particular expansion, it is correct to substitute this expansion inside a whatever partial differential equation?

the structure is uniform in z, so the solutions naturally separate as one function of x and y, times a function of only z.
the "naturally", typical of mathematical textbooks, usually hides considerations and theory that are obscure to me :)

Thank you for your patience. By the way, if you have in mind any book that can help me with these doubts I'll be grateful!

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...but I'm interested in this kind of theory applied to the electromagnetic context, in particular for resonant cavities....Can anyone suggest me a reference for this kind of treatment, oriented to the electromagnetism? The next step is to express these solutions into TE, TM and TEM solutions...

A mathematical justification of the modal expansion requires that you prove that the modes form a complete set.

A reference which proves completeness of modal expansions for both resonant cavities and waveguides in D.S. Jones' The Theory of Electromagnetism Chapter 4&5. I think you will find his discussion particularly lucid.

... However, I have my problems since I try to understand the reasons of hypothesis made in some treatments, while I feel more "safe" when I know that there is a math theory behind every assumption. Functional Analysis in the context of electromagnetism is probably where I should look for math justifications but it's really hard for me to find books that treat it applied in the field (many books treat the theory for scalar functions but very few also for vector functions, and usually in a manner that is not useful or easily understandable for me)....

Functional Analysis is not necessary in its entirety. Jones does introduce Lebesgue integration and Hilbert spaces but his exposition is very readable from an engineering electromagnetics viewpoint.

...A book that could be useful in this sense could be Operator Theory for Electromagnetics by Hanson, but I still have to check it properly...

I checked it for you. It does NOT prove completeness for vector fields only scalar fields are treated in depth.

EDIT: The resource I indicated show the proof for metallic waveguides.

vanhees71 and jasonRF
jasonRF
Gold Member
Thanks Deskswirl. I haven't really looked at Jones, beyond flipping through it once and seeing that it was not to be taken lightly!

Jason

jasonRF
Gold Member
That's the "so" part that is not immediate to me. I mean, I don't have the full background to notice particular structure of the coordinate system/boundaries and then apply these info to solve the partial diff equation.
If you have little or no experience applying separation of variables to solve boundary value problems in various coordinate systems, then I recommend you solve a few problems to get the hang of it. Since you already have a number of electromagnetics books, I would recommend starting with an undergrad level EM book that has boundary value problems and solving a few. Note that there are at least two things required for separation of variables to work 1) the PDE must be separable, and 2) the boundaries must occur along constant coordinate surfaces. Boundary conditions must also be such that the separation can occur, but this doesn't ever seem to be a problem in EM. Once you are comfortable with undergrad problems, solve some problems in your grad EM books. I don't think you need another book for this.

The general analysis in the beginning of Collin's chapter or in the reference you site separate the z-coordinate, but do not separate the two transverse coordinates. This is because of a) the $z$ dependence of $\nabla^2$ is simply $\partial_{zz}$ and the boundaries are at $z=\pm \infty$, so the $z$ dependence separates out, and b) they are not assuming that the cross-section of the structure is something "nice" like a rectangle, circle, ellipse, etc. that would be along constant coordinate surfaces for a properly chosen coordinate system. Hence they leave the two transverse coordinates coupled.

the "naturally", typical of mathematical textbooks, usually hides considerations and theory that are obscure to me :)
In this case (noticing z can be separated), it basically just comes down to understanding how separation of variables works, and then looking at the actual equations you are trying to solve and the boundaries that you have. It isn't very deep. Proving that the solution formed with the linear combination of the separated eigenfunctions is much harder ...

Good luck,

Jason

Thank you Deskswirl for the suggestions.
I never heard of D.S. Jones' "The Theory of Electromagnetism", only of its "Methods in electromagnetic wave propagation". Luckily there is a lot of preview with google books, so I'll start taking a look with it.

Thank you Jason for the mathematical insights. Onestly, I never took a serious class in PDE, even if in EM class instead there where a lot of PDE involved, so here's why my little experience. For what I understood it seems to me that, for what concerns PDE, there is a tight bond between possibility of variable separation, type of variable separation and boundary conditions. I'll try to investigate more about it, thank you.

Very last doubt that maybe can be answered with a yes or no only:
GENERAL FUNCTIONAL ANALYSIS DOUBT: when I know that an operator has an eigenvalue problem definition and can be written in a particular expansion, it is correct to substitute this expansion inside a whatever partial differential equation?
, like it seems to be done on pag 106 of the Milton-Schwinger. This will help me at least to close the main doubt of that particular development and also to learn a direct application of the spectral theory of operators (if the answer to the question is yes, of course ;) ). This use could be very interesting and also will increase my vision of the generalizations that could be done in mathematics, as it was very exciting to me discovering, many years ago, that it is possible to write every element of an Hilbert space using an eigenvector basis.

Thank you again so much!

Thanks Deskswirl. I haven't really looked at Jones, beyond flipping through it once and seeing that it was not to be taken lightly!

You are right. I'd need a great deal of patience and time to work through Jones completely (I haven't yet - only sections) regardless of the clarity. But it would definitely be worth it! He does not include any examples because the entire book is an example.

Thank you Deskswirl for the suggestions.
I never heard of D.S. Jones' "The Theory of Electromagnetism", only of its "Methods in electromagnetic wave propagation"...

I found Jones' Theory to be much more useful for understanding the mathematics of E&M than his Methods book. I have heard of another book by Jones: Acoustic and Electromagnetic Waves which I imagine is just as well done.

solanojedi:

If you have access to a library look for a copy of Moon and Spencer' Field Theory for Engineers or their Foundations of Electrodynamics (also now in Dover reprint). Either one covers the separation of variables technique in vector field to the extent that you'll need to understand. You might even start with Laplace's (scalar) equation using Byerly's An Elementary Treatise on Fourier's Series: and Spherical, Cylindrical, and Ellipsoidal Harmonics (in Dover, old editions in public domain?) which is very good on PDEs and focuses on problem solving, not theory.

****************************

Finally, in answer to your first original question (I had been meaning to respond for the last three months ) the solution to the vector wave equation for source free regions is shown in papers by Wilcox and Nisbit:

Calvin Wilcox, Debye Potentials, Indiana Univ. Math. J. 6 No. 2 (1957) 167–201
A. Nisbet, Hertzian Electromagnetic Potentials and Associated Gauge Transformations, Proc. R. Soc. Lond. A 1955 231 250-263.

and in regions with sources by Nesbit

A. Nisbet, Electromagnetic Potentials in a Heterogeneous Non-Conducting Medium, Proc. R. Soc. Lond. A 1957 240 375-381.

Both of Nisbet's papers are quite general in nature (and probably not very useful but it is reassuring to see that it has been shown). Wilcox only shows the spherical representation. Instead of finding the three eigenvector groups L, N and M (Morse and Feshbach) you mentioned in your original post Wilcox uses two scalar debye potentials. This seems to be a more direct route noting that those two scalar potentials multiplied by a pilot vector are related to two of the three L, N and M.

The best description of the scalar Debye (or equivalently Hertz, Mie, Bromwich, Whittaker, etc.) potential solutions to the vector wave equation in their engineering usage is given in Electromagnetic Fields by Van Bladel (2007) for the special time harmonic case (as suggested previously by Jason).

Also of note is Kerker's The Scattering of Light and Other Electromagnetic Radiation (1969) wherein chapter 3 he gives a table comparing notation and a more complete historical summary (if you are interested) than Nisbit.

Finally, I add this for posterity, the easiest demonstration (in spherical coordinates) of the vector harmonic functions in given in Chapter 4 of Bohren and Huffman's Absorption and Scattering of Light by Small Particles (1998).

Thank you Deskwirl, you gave me many reference, so now it'll take time (especially for me! :) ) to try to dig more in these arguments.

Thanks again everyone.

I'm sorry Deskwirl, if you have the Moon and Spencer' Field Theory for Engineers can you post the chapter index? I'm not able to find a copy in my library, so if the arguments are worth it I can think of buying on amazon.
Thanks again!

I posted the preface and contents on the amazon page.

https://www.amazon.com/dp/0442054890/?tag=pfamazon01-20

I'd say its worth the current price although I don't actually own a copy myself - I managed to get to the library before they closed! I DO however have a copy of their books on Vectors and PDEs. Both are very well done as is their Electrodynamics book and Field Theory Handbook which is absolutely one of the most convenient math references available for special functions in orthogonal coordinate systems. Can't say yet about the Holors book its on my reading list.

I find it interesting that many of the easiest to read and useful books in this area of applied math were published in the 1960s. Probably because it is no longer active directly as a research field.

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