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Consider a monochromatic plane wave (I am using bold to represent vectors) $$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, $$ $$ \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}. $$

There are a few ways to simplify this notation. We can use the complex field $$ \tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} $$ to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the complex amplitude ($\tilde{\mathbf{E}}_0$) when adding or manipulating fields.

However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity: $$ \tilde{\mathbf{E}}_0 e^{i\mathbf{k} \cdot \mathbf{r}} $$

My question is, what is this quantity called? I've been thinking time-averaged complex field, but then again, it's not really time-averaged, is it? Time-independent? Also, what is its notation? $\langle\tilde{\mathbf{E}}\rangle$?

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  • $\begingroup$ you should be careful with your notation. $\vec E_0$ and $\vec B_0$ could be complex themselves (e.g. for elliptic polarization), so your collection of $\vec E + i\vec B$ is in general not possible. But since $i\omega \vec B = -\textrm{curl} \vec E$ in the monochromatic case, you only need the E-field for for complete description $\endgroup$ Commented Nov 10, 2010 at 12:44
  • $\begingroup$ @Tobias Kienzler, Agreed. But the E and B-fields can still be gotten from the real and imaginary parts of $\tilde{\mathbf{E}}_0$, though. $\endgroup$
    – ptomato
    Commented Nov 10, 2010 at 15:11

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Stationary field or monochromatic field. Yes, basically that is the field including the $e^{i\omega t}$ term, but even when it is omitted one still knows what is meant.

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  • $\begingroup$ It's not a stationary (standing) wave though, importantly. $\endgroup$
    – Noldorin
    Commented Nov 10, 2010 at 13:14
  • $\begingroup$ @Noldorin: stationary does not necessarily mean constant, if only the complex phase changes but not the amplitude, "stationary" is still ok (at least in my environment) $\endgroup$ Commented Nov 10, 2010 at 13:16
  • $\begingroup$ @Tobias: That's not what I was saying. I'm simply referring to standing waves. $\endgroup$
    – Noldorin
    Commented Nov 10, 2010 at 21:23
  • $\begingroup$ @Noldorin: Agreed, I misunderstood you there $\endgroup$ Commented Nov 11, 2010 at 8:03
  • $\begingroup$ @Tobias: No prob. Physics certainly has a few 'overloaded' terms. And this question is really one of terminology, to confuse things further... $\endgroup$
    – Noldorin
    Commented Nov 12, 2010 at 4:01
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I'd call it the initial complex field since it's {E(t=0)}. No reference though.

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"Space-dependent phasor" makes some sense. Electrical engineers will often just say that's the "complex amplitude of the time-harmonic field" or "time-harmonic phasor".

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