On the eciency of the second harmonic generation in optical waveguides: TM case LIBORKOTAC Ï KAANDJIR Ï I Â C Ï TYROKY Â Institute of Radio Engineering and Electronics AS CR, 182 51 Praha 8, Czech Republic (E-mails: kotacka@ure.cas.cz, ctyroky@ure.cas.cz) Abstract. In this paper we analyse the second harmonic generation of TM polarisation in a planar waveguide with a non-linear anisotropic substrate using the coupled mode theory under the non-depleted pump approximation. It is an extention of our previous paper in which we treated TE polarisation. Moreover, both guided waves and C Ï erenkov radiation are now taken into account simultaneously. It is shownbothanalyticallyandnumericallythattheC Ï erenkoveciencypeakfollowsthe L 3=2 dependenceon the interaction length L also for TM polarisation. In an anisotropic KTP/Si 3 N 4 =SiO 2 waveguide, higher maximumattainablesecondharmonicgeneration(SHG)eciencywascalculatedfortheTMpolarisation than for the optimum TE case. Key words: C Ï erenkov regime, optical waveguides, second harmonic generation 1. Introduction Many papers devoted to the analysis of the second harmonic generation (SHG) in optical waveguides dealt with the more complicated case of TM polarisation. In the Tamada's paper (Tamada 1991), perhaps the most detailed study of the C Ï erenkov SHG in planar waveguides ever published, both the substrate and the guiding layer were supposed to be optically non- linear, but only one dominant non-linear optical coecient was considered in each material. The same presumption was made in the performance comparison of another con®guration for SHG in planar waveguides (Thyagarajan et al. 1993). Fluck et al. (1996) treated both polarizations in KNbO 3 channel waveguides, using a single eective non-linear coecient. Chang and Shaw (1998) studied TE±TM conversion using an eective non- linear coecient as well. Li et al. (1990) presented both theoretical and experimental study of the C Ï erenkov SHG in proton-exchanged LiNbO 3 using the complete tensor of non-linear coecients. In that work, the `classical' (large angle) C Ï erenkov SHG regime was considered. A very de- tailed theoretical analysis of C Ï erenkov-type SHG in slab waveguides was given by Hashizume et al. (1990). Their study contains an analysis of all possible polarisation conversions (TE±TE, TE±TM, TM±TE, and TM± TM) taking into account all pertinent non-linear coecients. The authors Optical and Quantum Electronics 33: 541±559, 2001. Ó 2001 Kluwer Academic Publishers. Printed in the Netherlands. 541