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!


Maybe not the perfect answer for your question, but one that is inspired by the practicality:

Because the conversion efficiency (not just the converted power directly) increases with increasing fundamental intensity, you will get more SH with higher intensity (as long as you stay under the damage threshold of your crystal/potential coatings of the crystal surface). So if you're limited to these two cases (no lens/this particular lens), try it out and see. Note that your input beam profile and focussing regime also influence the output beam profile and divergence, so depending on what you want to do with the SH afterwards you will need different collimation optics for the two different cases.

You could use the analytic expression as a boundary for your expectation of the conversion efficiency and test the validity with a measurement. You could also calculate, with your focussing parameters, what the deviation from a plane wave is at the end faces of your crystal, assuming the focus is in the centre, and see whether a plane wave approximation still makes sense.


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