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. Third harmonic generation (THG)^{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 |
Fig. 1.7.3.16 shows the three possible ways of achieving THG: a cascading interaction involving two χ^{(2)} processes, i.e. and , in two crystals or in the same crystal, and direct THG, which involves χ^{(3)}, i.e. .
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.
When the SFG conversion efficiency is sufficiently low in comparison with that of the SHG, it is possible to integrate the equations relative to SHG and those relative to SFG separately (Boulanger, Fejer et al., 1994). In order to compare this situation with the example taken for the previous case, we consider a type-I configuration of polarization for SHG. By assuming a perfect phase matching for SHG, the amplitude of the third harmonic field inside the crystal is (Boulanger, 1994)withΓ is as in (1.7.3.59).
(1.7.3.73) can be analytically integrated for undepleted pump SHG; , , and so we havewithwhere the integral J(L) is
For a nonzero SFG phase mismatch, ,
Therefore (1.7.3.75) according to (1.7.3.78) is equal to (1.7.3.72) with , and 100% transmission coefficients at ω and 2ω between the two crystals.
As for the cascading process, we consider a flat plane wave which propagates in a direction without walk-off. The integration of equations (1.7.3.24) over the crystal length L, with and in the undepleted pump approximation, leads to
According to (1.7.3.36) and (1.7.3.38), the integration of (1.7.3.79) over the cross section, which is the same for the four beams, leads towithwhere is in m^{2} V^{−2} and λ_{ω} is in m. The statistical factor is assumed to be equal to 1, which corresponds to a longitudinal single-mode laser.
The different types of phase matching and the associated relations and configurations of polarization are given in Table 1.7.3.2 by considering the SFG case with .
References
Boulanger, B. (1994). CNRS–NSF Report, Stanford University.Boulanger, B., Fejer, M. M., Blachman, R. & Bordui, P. F. (1994). Study of KTiOPO_{4} gray-tracking at 1064, 532 and 355 nm. Appl. Phys. Lett. 65(19), 2401–2403.
Qiu, P. & Penzkofer, A. (1988). Picosecond third-harmonic light generation in β-BaB_{2}O_{4}. Appl. Phys. B, 45, 225–236.