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We know that in free space, a propagating electromagnetic wave is always transverse. However, along a dielectric waveguide, the propagating wave can have longitudinal components. The exist of the longitudinal component makes the group index of refraction not equal to the phase group of refraction of the waveguide.

My questions is can one by using some combinations of propagating waves from any direction (even from the perpendicular direction to the waveguide) to make the field purely transverse or longitudinal in presence of a cylindrical fiber waveguide?

I am highlighting these two words here, and define the waveguide/optical fiber axis direction as the longitudinal direction and the plane perpendicular to the waveguide axis is the transverse plane. So, if a combined field oscillating along the fiber axis, I treat this as a longitudinal local field. Without this definition, one would get confused if we allow the light incident from any directions.

To be clear, meanwhile, I am talking about the electromagnetic field at least on one/periodical crossing section(s) of the propagating field or at least on a line, not just one or two specific/trivial points in space or the entire space. Thanks.

Xiaodong Qi
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2 Answers2

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Pure longitudinal modes, where either the electric or the magnetic field alone are wholly longitudinal (not needfully both vectors at once) are ruled out by Poynting's theorem. Such a wave could needfully propagate power only transversely to the guiding direction (by Poynting's theorem).

Pure TEM modes (i.e. no longitudinal component) with compact support can only exist on waveguides with two closed Jordan curves, one needfully inside the other, at each cross section. This means we have to have conductors for the electric field lines to end on, which means that they are practically ruled out for optical waveguides (although you could probably come close at longer wavelengths using metal coated waveguide technology). TEM modes of course exist in co-axial cables and microwave striplines at microwave frequencies.

The words with compact support are important, for a plane EM wave is of course a purely TEM mode.

To understand briefly why the TEM situation arises, it can be shown that TEM modes exist where the transverse field configurations of the electric and magnetic fields are exactly the same at each cross section as for electro- and magneto-static fields; see, for example, my answer here. Conversely, one can show a TEM field needfully has this analogy with static fields. But this means that if the field has compact support, then there is a zero potential equipotential formed by some closed Jordan curve that bounds the support region, i.e. a hollow conductor. The only solution to Laplace's equation, unless there are singularities within the boundary, with this behavior is a constant potential within the boundary, i.e. no electric of magnetic fields. So this means that there must be a singularity within or, equivalently, a second equipotential Jordan curve within. So we are talking about a strip line, with two different equipotential plates, or a generalized co-axial cable.

Selene Routley
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If your question is whether waveguides can support TEM (with transverse electric and magnetic field) waves, then the answer is generally no. There is a fundamental theorem that for a TEM wave the waveguide must be open. An example waveguide which supports TEM waves is the parallel plate waveguide. Closed waveguides such a circular or rectangular do not support TEM waves. [see Jin Au Kong "Electromagnetic Wave Theory", 1986, p. 196].

Note also since the waveguide modes are orthogonal, you cannot expect full cancellation of the transverse components (in the full cross section) by superposition of modes unless the modes have no transverse components in the first place.

This is generally not true for cancellation only in a part of the cross section. There probably one can achieve cancellation of the transverse components (I am mot fully sure there), but one generally would need infinitely many modes with excitation of a certain phase and amplitude.

Andreas H.
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