Lost Factors of 2 Around Gaussian Beams, Peak Intensities, Damage Thresholds, and Effective Mode Areas

Optical intensity is simply optical power divided by area, but what is the area of a Gaussian beam? People often just use ($\pi w^2$), where ($w$) is the Gaussian beam radius — the radius where the amplitude drops to ($1/e$) times its peak value, or the intensity to ($1/e^2$) times the peak value. So what intensity do we get when we just take ($P / \pi w^2$) — the peak intensity? No, the peak intensity is actually 2 times higher, and this is often missed.

In the context of light pulses with Gaussian spatial profile, we have the same issue concerning their peak fluence. In addition, one may wonder what the peak power is for a given pulse duration; this issue is different as pulse durations are usually given as FWHM values. My article on Gaussian pulses clarifies that.

A particularly nasty issue is in specifications concerning threshold intensities or fluence values for laser-induced damage. Often, people use Gaussian beams for such measurements and derive a damage threshold in W/cm² or J/cm² — without telling the reader how the intensity or fluence was calculated from the power or energy. Then you have to speculate whether they naively used the area ($\pi w^2$) or applied the extra factor 2. Unfortunately, that problem is not very rare in the literature. As a consequence, the true damage threshold is often 2 times higher than reported. At least, the problem may disappear if the user applies the same sloppiness as the author when applying the value! But we shouldn't bet on that.

As a diameter of a circle is two times larger than the radius, it would obviously be a good idea to clearly state which one is meant. But there are plenty of research papers where a “beam size” is specified, letting the reader speculate whether the radius or diameter was meant. Another chance to be off by a factor 2.

In fiber optics, we often use the concept of an effective mode area, which is defined with this equation:

Also, the transverse profile of the fundamental mode is often approximated with a Gaussian function. For a Gaussian beam with a beam radius as explained above, the mode area is ($A_\textrm{eff} = \pi w^2$).

Recently, I encountered amazing problems when getting my encyclopedia article on effective mode area checked with Gemini. It claimed that for a Gaussian beam my formula was wrong; it must be ($A_\textrm{eff} = \pi w^2 / 2$). But this is not true! Fortunately, I had not used a wrong formula in many places (also in numerous simulations) for many years! (I quickly confirmed my equation with a numerical test — which I often find to be the easiest way and less error-prone than calculating some integrals analytically.) But for curiosity, I researched more on this issue:

• I asked ChatGPT 5: “What is the effective mode area for a Gaussian beam with Gaussian beam radius w?” It also gave me that wrong result — nicely explained in detail with formulas, but in the last step (a simple division!) it made that stupid mistake. It corrected it only when being asked to check the last step again.
• I then asked Perplexity the same question, and it presented the same wrong result. Being asked whether it might be off by a factor 2, it told me that the difference comes down to whether you use the ($1/e$) radius for amplitude (which I do, as does basically everybody else) or intensity. But if you choose the latter, you get the area two times higher, not lower! It conceded that when I presented my argument. But how can the area then be ($\pi w^2$)? Perplexity presented essentially the same reasoning as ChatGPT did — with the same stupid mistake in the last step. Maybe Perplexity used ChatGPT internally — it won't reveal which large language model (LLM) was used.

It is amazing (a) that these LLMs give such good-looking answers even on complicated matters, and (b) that they can so easily fail in the simplest step, only requiring a division.

By the way, you might get correct answers when asking the same LLM some other day. It is just a lottery.

Keep in mind: The core mechanism of large language models is pattern recognition and next-token prediction, not stepwise, reliable logical or mathematical reasoning — even if you feel it must have used true reasoning to produce that output!

Still, it is very useful for me to do error checking of my encyclopedia articles with AI. It helps me to locate many (mostly small) deficiencies that I would not have found otherwise. But every single case needs to be checked carefully.

And your conclusion should be (just as from my previous AI test concerning photonics questions): Never trust AI, not even concerning such basic things like the effective area of a Gaussian. Instead, read an authoritative text, e.g. from the RP Photonics Encyclopedia. Certainly, this can also contain mistakes, but with a substantially lower probability — particularly when it comes to an online resource which is continuously improved by a diligent expert who cares about quality and reliability. Human expertise cannot be replaced by AI — at least not for now, and probably not any time soon.