International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 209-210
Section 1.7.3.3.3.1. SHG () and SFG () in different crystals^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
We consider the case of the situation in which the SHG is phase-matched with or without pump depletion and in which the sum-frequency generation (SFG) process (), phase-matched or not, is without pump depletion at and . All the waves are assumed to have a flat distribution given by (1.7.3.36) and the walk-off angles are nil, in order to simplify the calculations.
This configuration is the most frequently occurring case because it is unusual to get simultaneous phase matching of the two processes in a single crystal. The integration of equations (1.7.3.22) over Z for the SFG in the undepleted pump approximation with , and , followed by the integration over the cross section leads towithP^{ω}(L_{SHG}) and P^{2}^{ω}(L_{SHG}) are the fundamental and harmonic powers, respectively, at the exit of the first crystal. L_{SHG} and L_{SFG} are the lengths of the first and the second crystal, respectively. is the SFG phase mismatch. λ_{ω} is the fundamental wavelength. The units and other parameters are as defined in (1.7.3.42).
For type-II SHG, the fundamental waves are polarized in two orthogonal vibration planes, so only half of the fundamental power can be used for type-I, -II or -III SFG (), in contrast to type-I SHG (). In the latter case, and for type-I SFG, it is necessary to set the fundamental and second harmonic polarizations parallel.
The cascading conversion efficiency is calculated according to (1.7.3.61) and (1.7.3.70); the case of type-I SHG gives, for example,where Γ is as in (1.7.3.59).
(n^{ω}, T^{ω}) are relative to the phase-matched SHG crystal and () correspond to the SFG crystal.
In the undepleted pump approximation for SHG, (1.7.3.71) becomes (Qiu & Penzkofer, 1988)within W^{−2}, whereThe units are the same as in (1.7.3.42).
A more general case of SFG, where one of the two pump beams is depleted, is given in Section 1.7.3.3.4.
References
Qiu, P. & Penzkofer, A. (1988). Picosecond third-harmonic light generation in β-BaB_{2}O_{4}. Appl. Phys. B, 45, 225–236.