I have come across the following expression for the electric component of a laser field:

$$ \mathbf{E}(t) = \mathbf{E_0}e^{i\omega t} + \mathbf{E_{0}}^*e^{-i\omega t} $$

Where $\omega$ is the frequency of the laser, and $\bf{E_0}$ is the laser amplitude which is a complex vector.

Why is there two terms and what is the meaning of a complex amplitude? Why isn't the electric field given by:

$$ \mathbf{E}(t) =\mathrm{Re}(\mathbf{E_0} e^{i\omega t}) ? $$

  • 4
    $\begingroup$ Except for a factor of 2, your two expressions are the same. Taking a complex number and adding its conjugate makes it real. $\endgroup$ – Javier Nov 16 '16 at 20:34
  • $\begingroup$ Even if it weren't its complex conjugate there's no requirement for the electric field as solution to the Maxwell equations to be real: the observed quantity is in any case its module (which is however always a real number). $\endgroup$ – gented Nov 16 '16 at 22:34

The first expression gives the real (double) electric field as a function of time. Writing the complex field $\mathbf{E_0}e^{i\omega t}=E_0 e^{i\phi}e^{i\omega t}$ you get by using the Euler formula $$E_0e^{±i(\omega t+\phi)}=E_0[\cos{(\omega t +\phi)}±i\sin{(\omega t+\phi)]}$$ the real electric field time dependence with the amplitude $2E_0$, angular frequency $\omega$, and phase $\phi$ $$E(t) =\mathbf{E_0}e^{i\omega t} + \mathbf{E_{0}}^*e^{-i\omega t}=2E_0\cos{(\omega t +\phi)}$$ The formula $$E(t) =\mathrm{Re}(\mathbf{E_0} e^{i\omega t})=E_0\cos{(\omega t +\phi)}$$ gives the time dependence of the real electric field with the amplitude $E_0$. Thus both formulae give the real field time dependence but you hace divide the first by 2 to have the same real amplitude.


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