# How do I know whether waveguides support TE or TM modes?

I'm working through the formalism of waveguides, and I've seen many different situations with various boundary conditions: hollow metal rectangular waveguide, dielectric slab, dielectric rectangular waveguide, dielectric circular waveguide. Some of these support TE and TM modes and some don't. For the channel waveguides (rectangular and circular, not the slab ones), it seems like a good rule of thumb is this: you can get TE and TM modes if your waveguide is a conductor, but if your waveguide is a dielectric surrounded by another dielectric, you don't get TE and TM modes.

My question is: what is the intuition behind this last sentence?

Part of the difficulty is that many treatments of the situations where there are TE/TM modes simply say "let's set $$E_z=0$$ or $$H_z=0$$." And then they don't say that in the dielectric cases, and I don't know what stops me from doing that in the dielectric case.

Some math will undoubtedly be helpful, but I'm really looking for the intuition as well.

That's my basic question: how can I know intuitively when a waveguide supports TE/TM modes, or what stops me from simply setting (one of) the longitudinal field components to zero in a dielectric waveguide?

Edited for clarity.

• Can you please edit the question to clarify the section beginning 'For the waveguides that confine in two dimensions ' ; is that two sentences or one?
– CWPP
Commented Aug 17, 2022 at 4:53
• I suspect that simple TE/TM thinking is not applicable to the dielectric slab problem. The propagation speed of an electromagnetic wave in space is $c_0=1/\sqrt{\epsilon_0\mu_0}$, and its speed in dielectrics is $c_1=1/\sqrt{\epsilon \mu_0}$. The speed differences can be a source of difficulty in obtaining analytical results for dielectric slab problems. Commented Aug 17, 2022 at 5:54
• @CWPP I've edited; does it make more sense? Thanks for the attention to the question. Commented Aug 17, 2022 at 11:10
• @HEMMI Although the waves travel at different speeds, I think you can still get TE/TM modes in the dielectric slab case. (I'm looking at Photonics by Yariv and Yeh where they work out the problem in Secs. 3.1 and 3.2.) I think it has something to do with the boundary conditions, but if I understood what, I wouldn't be asking the question. Commented Aug 17, 2022 at 11:12
• I think the edit is very good (I've upvoted the question) and I'm thinking about it. It clearly comes from the boundary conditions but can we understand how in conceptual terms?
– CWPP
Commented Aug 17, 2022 at 14:55

In a longitudinally homogeneous waveguide if it is filled uniformly with linear dielectric or paramagnetic material you will get TE and TM modes, as you can find it in any book on waveguides. If you maintain the longitudinal homogeneity but transversally the permittivity and/or permeability may vary then the propagating wave must have a longitudinal component. This you can see if you assume piecewise constant filling and try to match the fields tangential/normal to the jump. More formally you can derive it by using the so-called Marcuvitz-Felsen equations, see Section 2.2a in Felsen-Marcuvitz. What is interesting about this is that you obtain a pair of vector differential equations (transmission line equations) for the 4 transversal components without the longitudinals showing up the longitudinal components being proportional to the to the transversal curls. That you can reduce the original six-variable Maxwell's equation to four unknowns can be expected since we know that the ME can equally be represented by a scalar (1) and a vector (3) potential, altogether 1+3 = 4 unknowns.

Interestingly, and rather unexpectedly, at the discrete frequencies at which the propagating wave stops propagating, i.e, at cut-off, but not below where the mode is evanescent, the longitudinal field components may become zero. You can prove that to yourself, not too difficult, using the same Marcuvitz-Felsen equations, and check the components at cut-off. (If you need help how to do this with the MF equations then ask a separate question and I will try to respond to it.)

• Thanks for the answer. Your first three sentences are exactly what I'm trying to gain a physical (not mathematical) intuition for. Commented Sep 16, 2022 at 16:13
• Do you have any physical intuition why a homogeneously filled waveguide has TE/TM modes? Do you have physical intuition why free space is TEM modes? If yes, what is that physical intuition? Commented Sep 16, 2022 at 17:39
• That's a good point — I don't particularly have any intuition for these situations, and maybe I should ask a broader question. I understand that Maxwell's equations imply certain things, e.g. in free space $\nabla \cdot E = \nabla \cdot B = 0$ and for a monochromatic place wave this means $k \cdot E = k \cdot B = 0$, and conversely if you have sources or if you don't have a plane wave you might not get this, but I guess this isn't exactly a physical intuition. Commented Sep 17, 2022 at 14:02
• and that is a wrong intuition in general, for example, the near field that is "non-propagating" of an antenna has all 6 components while divE=divB=0. Even in the far-field (propagating) transversality holds only in an asymptotic sense, see radial waves. Commented Sep 17, 2022 at 14:14
• Yes, this case is covered by the qualifications and exceptions mentioned in my last comment ("monochromatic plane wave", for example). Commented Sep 17, 2022 at 15:41

I cannot be sure the following is relevant, and you don't consider some specific waveguide, so let me discuss a situation I am familiar with: diffraction of a (cylindrical) electromagnetic wave on a circular cylinder. This problem was solved exactly by Wait.

It turns out that for a perfect conductor an incident TM (TE) wave creates a refracted TM (TE) wave, however this is not true for a dielectric and oblique incidence, so, for example, an incident TM wave creates both refracted TM and TE waves. So it is possible that in some cases there just cannot be pure TM or TE guided waves in a waveguide, depending on the electric properties of the latter.

EDIT (Sept 16, 2022) : Some confirmation of the above from Journal of the Optical Society of America Vol. 51, Issue 5, pp. 491-498 (1961):

In a metallic guide there are two sets of solutions, the transverse electric and transverse magnetic modes. In the dielectric guide all but the cylindrically symmetric modes TE$$_{\mathrm{om}}$$ and TM$$_{\mathrm{om}}$$ are hybrid; i.e., they have both electric and magnetic $$z$$ components.

I suspect something similar takes place for rectangular waveguides.

• Thanks for your answer. I've previously seen the idea of propagating modes in optical fibers as waves bouncing back and forth and creating transverse standing waves. Your answer might relate to that in a useful way. Is that the direction you were going for, or did I misunderstand? Commented Sep 16, 2022 at 16:15
• @flevinBombastus : Not quite. Please see my EDIT. Commented Sep 17, 2022 at 3:02
• Right. Something similar does take place for rectangular waveguides, and I'm just trying to gain some physical intuition for why all of this is true. Commented Sep 17, 2022 at 13:51