Equation of an electric field of a Gaussian pulse

Question

I am trying to write the equation of electric field of a Gaussian pulse of light to do some MATLAB analysis and other analysis. What I want is an electric field $E(y,t)$ of a Gaussian pulse traveling with a cosine function along $y$. This is the equation I have:

$$ E(y,t) = \cos(\omega t-ky)\exp[(-(t/tp)^2) + j\omega t] $$

In addition, in this situation, I imagine the Gaussian function will produce the envelope with Gaussian width and time tp and the cosine function will be the carrier signal. Are my assumptions correct?

Answer 1

Not quite.

• You have two occurrences of $\omega t$, which is obviously not going to do what you want.
• The envelope needs to travel with the pulse, and the way you've put it it's just a global factor that goes up and down with time over the entire line.

Other than that, you're basically correct. The pulse you want to use is probably something like \begin{align} E(y,t) & = \mathrm{Re}\mathopen{}\left[e^{i(ky-\omega t)} e^{-{(t-y/c)^2}/{\tau^2}} \right]\mathclose{} \\ & = \cos(ky-\omega t)e^{-{(t-y/c)^2}/{\tau^2}}. \end{align}

Depending on your application, the carrier-envelope phase may be somewhere between negligible and absolutely crucial. If it does matter, then it may be necessary for you to modify the pulse to $$ E(y,t) = \cos(ky-\omega t-\varphi_\mathrm{ce})e^{-{(t-y/c)^2}/{\tau^2}}, $$ with a relative phase of $\varphi_\mathrm{ce}$ between the peak of the envelope and the nearest maximum of the carrier.