Beam Quality in Second-Harmonic Generation

Posted on 2008-01-27. Permanent link: http://www.rp-photonics.com/spotlight_2008_01_27.html

Author: Dr. Rüdiger Paschotta, RP Photonics Consulting GmbH

Ref.: encyclopedia article on frequency doubling and beam quality; spotlight 2007-10-17

A question from a colleague triggered me to discuss the relation between beam quality of a pump beam and the frequency-doubled beam generated in a nonlinear crystal, as this has some instructive aspects. I distinguish two different cases:

Frequency Doubling with Gaussian Beams

When the pump beam is a Gaussian beam, the second-harmonic beam will have a somewhat reduced beam radius, as explained in the Photonics Spotlight of 2007-10-17. Essentially this is because the nonlinear polarization, which is proportional to the square of the pump amplitude, is narrower than the pump. Due to the smaller beam radius at all locations, the beam divergence is also reduced. If there is weak pump depletion and no spatial walk-off, and the focusing is not very strong, the beam area in the focus will be half that of the pump beam, and the divergence will be smaller by a factor square root of two. The beam parameter product is thus maintained, and the M² factor remains unity.

With some spatial walk-off, the second-harmonic beam becomes wider, but the divergence angle is also reduced. Therefore, the beam quality does not need to be any worse.

The Multimode Case

For simplicity, we assume an approximately rectangular pump beam shape with many modes involved. In that case, we cannot expect that the frequency-doubled beam will be narrower. Its divergence will also be similar to that of the pump beam, so the beam parameter product will be similar, despite the doubled frequency. Therefore, the M² factor will be approximately doubled.

It may be surprising that the beam quality in terms of M² is degraded in that case, while it is preserved in the case of a Gaussian pump beam. This can be understood by considering the involved propagation modes. Every mode of the pump field will generate a corresponding frequency-doubled mode. In addition, sum frequency generation will generate modes with intermediate frequencies. In effect, the total number of modes in the second-harmonic field (considering just one dimension) is about doubled, and this explains the doubled M² factor.