-1
$\begingroup$

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:

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

$\endgroup$
  • 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.