# Can a wave's Poynting vector be in the opposite direction compared to its direction of propagation?

Can a wave's Poynting vector be in the opposite direction compared to its direction of propagation, and if so, what physical implications does it have?

As I understand, the poynting vector s can be represented as: s = E x H, both E and H being vectors, and since E and H are always perpendicular, they will form 90º and then the cross product will always be maximum and positive, but since I have been asked what the implications of the poynting vector having the opposite direction are, I am not sure.

• How would you define a positive or negative vector? A vector can point into any direction in space (opposed to numbers on the number line which can be left or right of the zero). – Photon Jan 14 at 9:16
• Depends on your coordinate system. If $\vec E=\hat i$ & $\vec H=\hat k$ then $\vec S=-\hat j$ – harshit54 Jan 14 at 9:25
• I define positive as in the same direction as the wave´s propagation direction – Jesus Chueca Jan 14 at 9:41
• @JesusChueca: If someone asks you for a clarification on a question, please edit the question rather than providing the clarification in a comment. I've edited it for you. – Ben Crowell Jan 14 at 14:21
• @harshit54: No, even if you fix a coordinate system, there is no way to define positive and negative vectors, only positive and negative components. – Ben Crowell Jan 14 at 14:21

In general the Poynting vector is the in the direction in which the wave propagates. Usually, for university level work etc, for the most part we already know the direction of the EM wave. Thus the Poynting vector comes out positive, and in the direction we have predefined. However, if, for instance, we rotate our coordinates, the Poynting vector will point in the direction of propagation in the new coordinate system.

Now the direction of propagation can for instance be in the direction of the negative z axis, which would yield a 'negative' vector.

Having said that, if by 'negative' one means that the amplitude is negative, i.e. the energy propagation is negative, this would imply the 'source' of the wave is actually gaining energy as opposed to radiating it. This is possible for instance when you have an antenna that receives radiation. What is meant by that is that the external wave is oscillating the charge in the antenna, thus deposits energy into the antenna.

In that case though you need to take the viewpoint of the charge at the antenna to get the so-called negative Poynting vector.

• Yes, by negative I mean that the poynting vector is in the opposite direction of the wave. So I understand from your reply that the poynting vector will ALWAYS be in the same direction of the wave´s propagation, and that it can´t be opposed to it. – Jesus Chueca Jan 14 at 9:39
• @JesusChueca Actually the Poynting vector defines the direction of propagation. Think about it: Is there another way to define direction of propagation, than by the direction of energy flow, which the Poynting vector specifies? – Andreas H. Jan 14 at 9:52
• I understand what you mean... and I do agree with you that the direction of propagation is the direction of energy flow, but in an electromagnetic wave where E and H are both always positive or negative at the same time, would it be possible to have a Poynting vector in the opposite sense of the propagation? – Jesus Chueca Jan 14 at 9:56
• @Andreas H. the direction of propagation (wave vector) is not always the same direction as that of Poynting's vector. In some crystals and plastics (calcite, polystyrene), there can be EM wave propagating in one direction but Poynting vector pointing in a different direction. – Ján Lalinský Jan 14 at 13:24

In terms of the Fourier transform $$\sum_{\vec{k}}A_{\vec{k}}\text{e}^{i{\vec{k}\cdot\vec{x}}}$$, a wave from some sorce is a superposition of plane waves with different wave vectors $$\vec{k}$$, including the opposite directions too but with different amplitudes and phases, especially if $$\vec{x}$$ is "close" to the source. At the same time the Poynting's vector $$\propto \vec{E}(\vec{x})\times\vec{B}(\vec{x})$$ at a given point of space $$\vec{x}$$ is unique.

Each plane (running) wave has the propagation direction along its wave vector $$\vec{k}$$, so does the wave energy flow. But generally the EMF is not a plane wave, it "spreads" or "focuses" thanks to the presence of different spatial harmonics in its Fourier decomposition.

In negative index materials (where $$\varepsilon$$ and $$\mu$$ are both negative) the Poynting vector $$S$$ is antiparallel to $$k$$. Such negative index materials have been realized with metamaterials where the effective electric permittivity and effective magnetic permeability are both negative.