The World record of highest efficiency of SHG was obtained experimentally, if I remember correctly, with low divergent super-Gaussian beam. The value is of order of 92%.....But all depends on power and radiation stiffness of crystal. Resistant crystal allows high intensity, and in this case divergence of radiation can be of minor importance. By the way, coherent effects allow conversion efficiency very close to 100%, since they allow using very high effective nonlinearity. 

Dear Prof. Aleksandr Aleksandrovsky,

 Thank you so much for your quick response. Can you please evaluate efficiency of second harmonic generation with zeroth order bessel beam in uniaxial crystal in critical phase matching mode (phase matching of a nonlinear interaction by adjustment of a propagation direction) ? And compare to that of gaussian beam.

No matter what the beam profile is. The spatial coherence and incident intensity and Peak Power are important to obtain efficient SHG.

As a practical matter, ~ 70% conversion efficiency for pulsed beams is often realizable given sufficiently narrow line width and suitable nonlinear crystals. There are wavelength regions that are very hard to obtain high conversion efficiencies (Many in the UV) due to unfavorable phase matching parameters and other types of problems like low crystal damage threshold as well as linear and nonlinear absorption of the generated beams. If you don't care about polarization there is a method called quadrature which utilizing two doubling crystals with suitable waveplates between the crystals.  This can boost efficiencies as high as 85%-95% but again doesn't work as well for doubling into the UV for reasons mentioned before.  If your beams have terrible divergence (or M^2) or are broadband (>1 nm), realizing high SHG conversion may not be possible.

As far as your original question about Bessel vs Gaussian beams unless the nonlinear crystal of interest has a very poor angular acceptance bandwidth, the differences of conversion efficiency between the two is probably negligible.  Of course this might not be the case for highly focused beams or with crystals whose walk-off Poynting vector is large ( BBO often has very high walk-off ).

Please note....even with very good beams.....there will be an optimum length crystal to obtain high conversion efficiencies.  If your crystal gets too long....you can have back conversion to the pump beam. 

Dear Lam Thanh Nguyen

I think your physical reasoning is right in general.

However, I think that the beam profile is not critical consideration factor in the SHG, and the important factors are the phase matching and the overlap integral between the fundamental and second harmonic waves. For the SHG in a bulk crystal, the role of overlap integral does not exist, and the SHG conversion efficiency is affected by the amplitude of incident fundamental wave with the phase matching wave vector. In general, the SHG by the interaction between two waves with non-phase matching wave vectors is negligible, and the low overlap integral and incident beam power could reduce the conversion efficiency of SHG.

Firstly, the SHG in a bulk crystal is assumed. If the amplitude of the plane wave component with the phase matching wave vector of Gaussian beam is larger than that of Bessel beam for the same pumping power, then the conversion efficiency by Gaussian is greater than that of Bessel beam, and Gaussian beam is better than Bessel beam for the second harmonic generation.

Secondary, we assume a case of the SHG by the interaction of fundamental guided wave and second harmonic guided wave in a planar waveguide. The field profile of fundamental guided mode is close to Gaussian distribution and the pumping Gaussian beam is highly coupled to the fundamental guided mode in the waveguide. The conversion efficiency by Gaussian beam could be greater than that of Bessel beam, and Gaussian beam is better than Bessel beam for the second harmonic generation. The focused Gaussian beam is used typically as the pump wave for the SHG.

For a case of SHG in the waveguides, the SHG conversion efficiency is also affected by the overlap integral between the mode of incident fundamental wave and the mode of second harmonic wave with the phase matching propagation constants.

In conclusion and in my opinion, the best beam for second harmonic generation is the beam profile which gives the largest amplitude of the incident fundamental wave of the phase matching wave vector(in a bulk crystal) or the largest coupling into the fundamental guided mode(in a waveguide).

From nonlinear optical theory, the best beam is the one whose total overlap is greatest. SHG considers a beam as two beams, and the NLO system must be oriented to separate it that way -- usually by polarization.  Then the two polarizations usually propagate in slightly different directions (which is one reason to use periodically-poled NLO material).  This would tend to recommend the Bessel beam, which doesn't diverge much and generally leads to more spatial overlap.  But the frequency overlap is important as well, and it is easier to get a narrow linewidth from a Gaussian beam.

The short answer is that the best beam depends on the circumstances; the reality is that virtually all SHG systems use Gaussian beams because they are so much easier to generate and can have the narrow linewidth needed.

I would like to thank Professor Aleksandr Aleksandrovsky, Professor Parviz Parvin, Professor Waverly Marsh, Professor Zung-Shik Suh, Professor Russell Marc Kurtz for these great discussions. And I will be very happy if other experts in nonlinear optics give more their opinions on this topic.