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I have the time dependence of the electric field $A(t)$ written in terms of the amplitude $E(t)$ and phase $\phi(t)$ as

$$A(t) = E(t) \cos[\nu t + \phi(t)].$$

I am told that this varies slowly in an optical period $2\pi/\nu$.

I am then told that

$$\delta(t) = E(t) e^{-i[\nu t + \phi(t)]}$$

is the "positive frequency part of the electric field." I understand that this is using Euler's formula, but I don't understand what is meant by the "positive frequency part of the electric field." How does $\delta(t) = E(t) e^{-i[\nu t + \phi(t)]}$ represent the "positive frequency part of the electric field" (and therefore what would be the "negative frequency part of the electric field")? Why is this distinction relevant/material in physics (in the study of electromagnetism/electrodynamics)?

The Pointer
  • 939

2 Answers2

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$$A(t) = E(t) \cos[\nu t + \phi(t)]=\frac{1}{2}E(t)[e^{j(\nu t + \phi(t))}+e^{-j(\nu t + \phi(t))}]=\frac{1}{2}[\delta(t)+\delta^{*}(t)]=\frac{1}{2}E(t)[e^{j(\nu t + \phi(t))}+e^{j(-\nu t - \phi(t))}]$$

Note that since both $\nu$ and $-\nu$ are angular frequency values present in the formulation, for any $\nu\neq 0$ the electric field includes a positive frequency and a negative frequency term. I know this distinction is important in the context of wave propagation where a standing wave may be interpreted as the superposition of two propagating waves, one going to the left and the other going to the right.

Newbie
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Obviously there is a physical thing that is oscillating as a sinusoid. The positive and negative exponentials are just a way of conveniently writing this. There is no information in the "positive frequency" component that isn't in the negative.

rogerkoulitt
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