Why is the $\cos^2$ envelope much more popular than $\cos^4$ to approximate the shape of a Gaussian laser pulse?

Question

The Gaussian laser pulse $$ E(t) = E_0 e^{\frac{-2\ln 2}{\tau^2} t^2} \cos{\omega t} $$ (where $\tau$ is the intensity FWHM) is often troublesome in numerical simulations due to the infinite temporal length of it's envelope, which can introduce artifacts unless a very wide temporal window is used. A widespread solution is to use a $\cos^2$ temporal profile, $$ E(t) = \begin{cases} E_0 \cos^\boldsymbol{2}\left(\pi \frac{t}{T_p}\right) \cos{\omega t} & \mathrm{if}\ -\frac{T_p}{2} < t < \frac{T_p}{2} \\ 0 & \mathrm{othervise} \end{cases} $$ with $T_p \approx 2.741 \tau$ being the full zero-to-zero length of the pulse. This envelope smoothly goes to zero at $t=\pm T_p/2$ and stays zero outside these bounds, making it suitable for such simulations.

What puzzles me is it's popularity over a $\cos^4$ shaped envelope,

$$ E(t) = \begin{cases} E_0 \cos^\boldsymbol{4}\left(\pi \frac{t}{T_p}\right) \cos{\omega t} & \mathrm{if}\ -\frac{T_p}{2} < t < \frac{T_p}{2} \\ 0 & \mathrm{othervise} \end{cases} $$

which has a shape much closer to a Gaussian envelope. In fact, it is so close it's barely distinguishable:

dashed: envelope, continous: field

However, publications seem to almost exclusively use $\cos^2$ for any simulational work involving ultrashort laser pulses. In fact, I found only 1-2 works that considered both $\cos^4$ and $\cos^2$, and I'm yet to encounter a work that exclusively chose $\cos^4$ over $\cos^2$. This is very uintuitive for me - I'd assume that the closer you are to the Gaussian profile, the better.

What is the reason? Why does $\cos^2$ have an almost exlcusive preference in simulational works, despite it's shape much farther from a Gaussian shape than that of $\cos^4$?

Answer 1

The best way to answer this question is to take a number of papers that explicitly discuss the choice of $\cos^2$ versus Gaussian envelopes, and to repeat their analyses using the Gaussian envelope (with cutoffs for various multiples of $\tau$), the standard $\cos^2$ envelope, and your $\cos^4$ envelope. Older papers tend to be more explicit about these decisions than newer papers, so dredging the literature may help you here.

There are a number of possible explanations you might discover:

1. Cultural inertia. Everybody uses $\cos^2$ because everybody else uses $\cos^2$.

2. A physical reason. Perhaps, as much as we love to find Gaussians emerging from statistical processes, the Gaussian envelope is (or was) actually a poorer fit to the physical laser pulses in some papers. For example, if you have a continuous beam through two crossed linear polarizers and generate a pulse train by spinning one polarizer at constant speed, your pulse train will have envelope $\cos^2$. (Yes, I am aware this is not how ultra-fast pulses are generated.)

3. A Fourier-analysis reason. To quote a comment under your question:

   I wonder if the reason is the following: you only need $2\Omega$ (and a constant) to represent a $\cos^2\Omega t$ but also $4\Omega$ in your case. After modulation you have $\omega$, $2\Omega\pm\omega$, $4\Omega\pm\omega$. All those frequencies do not have in general rational ratios so discretizing them may produce some artifacts (don't ask me what:P) when you propagate them at very large times.

4. An analytic-continuation reason. The envelopes of $\cos^2$ and $\cos^4$ have the same value and first derivative at the cutoff, but the quartic has larger values for the higher-order derivatives. (Well, the nonzero higher-order derivatives.)

5. A stupid reason. Your question refers to "the zero-to-zero time" as if that were a thing which exists. But for a "true Gaussian" pulse, it doesn't: the envelope is never zero. Perhaps the unacceptable artifacts arise from truncating the Gaussian in the neighborhood of your choice of $2.71\tau$, while extending the analysis to $5\tau$ or $6\tau$ makes the artifacts negligible.

6. A signal-to-noise reason. Perhaps the Gaussian envelope needs to go out to $6\tau$, but real experiments have noise in the data out that far, so such analyses wind up focused on the background.

7. A boiling-frog reason. ("If you drop a frog in boiling water it will panic and jump out, but if you put the frog in tepid water and boil it slowly, the frog won't notice until it's been cooked," is the proverb.) Your sample pulse, with five or seven maxima within the envelope, has frequency $f≈1/\tau$. Perhaps earlier in the literature, when a "short" pulse had $\tau\sim100/f$, the artifact structure was different, but a better choice is appropriate in the "ultra-short" era.

I can imagine a careful answer to your question being written up for publication in a journal like Am.J.Phys. — which might actually be a good place to look for an answer, before you decided to write such a paper yourself.

Answer 2

cos^4 might seem intuitively appealing due to its Gaussian-like shape, the practical advantages of cos^2, particularly its computational efficiency, narrower bandwidth, and established practices within the scientific community, make it the preferred choice in many ultrafast laser simulations , important to remember that the choice of pulse envelope is often driven by a careful consideration of the specific requirements and trade-offs involved in each simulation scenario.