Second harmonic generation with strictly monochromatic light is a well-known problem of physics, whose solution is found in any textbook on nonlinear optics. However, nowadays, in several situations short-pulsed laser radiation is required [1] (for instance, for achieving high-intensity fields). In this regime, even for fields which are relatively weak, second harmonic generation is not a completely understood process, since the propagation equations of the fundamental and harmonic waves are no longer analytically solvable. Physically, this implies an incomplete understanding of effects which start to play a role in this pulse-length region, such as the influence of the group velocity mismatch between fundamental and harmonic waves, the energy transfer between both waves and time-dependent effects which may influence the conversion efficiency and the pulse shapes of fundamental and harmonic radiation.
We show that, under the assumption that both waves are amplitude-modulated, and that the slowly-varying-envelope approximation is valid, the propagation equations of fundamental and harmonic waves can be reduced to an intial value problem. Within this context, we derive a general analytical solution for an initial pulse of arbitrary shape, using the connection between second harmonic generation and the Liouville equation [2] established in [3]. The inverse problem, i.e., to determine the input pulse shape, given an asymptotic second harmonic pulse, is also solvable.
- My contributions to the subject
- My collaborators: H. Steudel, M.G.A. Paris, A. Kamchatnov and O. Steuernagel
| [1] See e.g. Y.R. Shen, "The Principles of Nonlinear Optics", Sect. 7.6 ( Wiley & Sons, New York, 1984 ); S.A. Akhmanov, V.A. Vysloukh and A.S. Chirkin, " Optics of Femtosecond Laser Pulses" ( American Institute of Physics, New York, 1992) | |
| [2] J. Liouville, J. Math. Pure Appl. 18, 71 (1853). | |
| [3] F.G. Bass and V.G. Sinitsyn, Ukrain. Fis. Shourn. 17, 124 (1972) (in Russian). |