# Focusing a Gaussian beam into a nonlinear crystal

Boyd analyzes in his book (Boyd's book), section 2.10.3, the case of harmonic generation using a focused Gaussian beam and he mentions that an analytic solution can be evaluated for certain special cases as the plane-wave limit (confocal length $$\,b\,$$ is much larger than the crystal's length $$\,L$$) as well as the tightly focused wave in the crystal (confocal length is much smaller than the crystal's length).

I'm trying to understand when do such limits hold and find the SHG intensity since I'm designing an experiment to produce SHG with an Argon laser ($$\lambda=488$$ nm) of $$P=800$$ mW and waist $$W_0=0.75$$ mm using a $$L=3$$ mm KDP crystal in type I phase matching ($$ooe$$), whilst assuming that the indices of refraction are $$\,n=1.5\,$$ (non-critical phase matching), and I have a $$f=250$$ mm lens I can use.

Without using the lens, I am indeed in the limit where $$b\gg L$$ and thus the plane-wave limit holds as expected. However, using the lens I have, the new waist of the beam focused into the crystal becomes $$W_0^\prime=\frac{\lambda f}{\pi W_0}\approx51.78\ \mathrm{\mu m},$$ and the new confocal length is $$b=\frac{2\pi}{\lambda}W_0^{\prime 2}\approx 34.5\ \mathrm{mm}$$ which is still larger than $$L$$. However, is it "much larger" ($$b>L$$ but is it considered also $$b\gg L$$? It is afterall about 11.5 times larger)? Does the plane-wave limit still hold? Can I still use the analytical solution known for plane-waves for the second harmonic intensity?

Any help would be much appreciated. Thanks!