# Self-focusing of laser beam

I have exam of „Nonlinear Optics“ tomorrow and I don't understand one thing. In the book it is said, that self-focusing occurs when power of beam reaches so called critical focusing power $$P_{cr}≈\frac{λ^2}{8n_0n_2}$$. But self-focusing occurs because of Kerr effect, when intense laser pulse induces refractive index change, since $$n=n_0 + n_2I$$. So shouldn't it be critical intensity, not critical power? Because intensity is power per unit area, and if we expand our beam, we distribute power to larger area - beam intensity drops. This is done in high power laser systems, where beams are being expanded in space so self-focusing won't occur and damage optical elements.

• What is the context where the $P_{cr}$ formula was introduced? Is there some implied spot size? For example if they're talking about what happens at the focal point of some converging lens, then they might have approximated the spot size as $\lambda^2$ Commented Jan 12, 2021 at 21:00
• Yes, there is spot size implied at first. So you want to say that this critical power is for fixed beam spot size? But why "critical power" is used? If we know fixed beam spot size and energy for self-focusing for that beam size, we can divide energy by spot size and get critical intensity.. Commented Jan 13, 2021 at 9:05
• It would be better if you posted your edit as an answer. Commented Jan 14, 2021 at 14:40
• Yes. Sorry, I am quite new to the stackexchange forums. I thought it's not appropriate to answer your own question. Commented Jan 14, 2021 at 18:57

So, I found it out. Before the exam start, I asked my professor this question. If a short laser pulse has equal or more power than so called critical focusing power, sooner or later beam will self-focus, but the distance of beam travel, until it self-focuses, can be very huge, and this distance is strongly beam size dependent. If we expand our beam waist radius $$w_0$$, the so called non-linear focal length $$z_{nf}$$ increases:
$$z_{nf} = \frac{2n_0w_0^2}{λ}\frac{1}{\sqrt{P/P_{cr}-1}}$$
So, for example, let's say that our pulse center wavelength is λ = 800 nm, it's power $$P = 2P_{cr}$$, and beam waist is 1 cm. That beam will self-focus after 25 km in air, and if beam is propagating in, for example, glass with refractive index about ~1.5, non-linear focal length will be 37.5 km.