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I know that the electric field component of electromagnetic wave should be written as the first equation shown, but some times it can be expressed in terms of complex amplitude of electric field as in second equation. I later found integration in the exponent term and I didn't know why this term appeared. Which one is correct, or more general?

$$ \mathbf{E}(\mathbf{r},t)=\mathrm{Re}\left[\hat{\mathbf e}F(x,y) A(z,t)\exp(i\beta_0-i\omega_0t)\right], $$ $$ \mathbf{E}=\exp\left(-j\int_0^z\beta_\text{eq}\mathrm dz\right)\mathbf{A}. $$

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  • $\begingroup$ I think you might need to clarify what $\bf{A}$ represents in the second, and why there is no time dependence there. Otherwise it looks like you are comparing apples and oranges. There's a missing $z$ in the first, by the way. The second appears to represent the accumulated phase while propagating through an inhomogeneous medium? What is $\beta_{\mathrm{eq}}$? It appears to be a function of $z$; is that correct? $\endgroup$
    – garyp
    Apr 8, 2014 at 20:30
  • $\begingroup$ A is the amplitude of electric field , yes there is a missing z in 1st equation,what is meaning by accumulated phase and why there is no accumulated phase in the 1st equation , βeq it represent equivalent propagation constant like β0 but also can depend on other terms if there is variation in propagation constant according to medium , βeq i don't know if it is depend on z or not i also wondering !! $\endgroup$
    – Mai Fouad
    Apr 8, 2014 at 22:13
  • $\begingroup$ The first is more or less valid for a homogenous medium. But now, looking at it more closely, I'm not certain it's a solution to the wave equation. It might have to be $F(x,y,z)$. Well, I'm not sure. At any rate, I don't see the connection to the second formula. Where did you find these equations? $\endgroup$
    – garyp
    Apr 8, 2014 at 23:21

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