2011-8-29Department of PhysicsSoutheast UniversityLecture eight: Phase matching and calculation of effective d1. Phase matchingaccording towhere(1) (2) (3) For nonlinear mixing processes that are sufficiently efficient to lead to depletion of the input beams, the functional dependence of the efficiency of the process on the phase mismatch is no longer given by Eq. (2). (4) As a result, the condition for perfect phase matching with collinear beams,But this is not common in use since the anomalous dispersion is always accompanied by the strong absorption effect which hinders a clear observation of three-wave interaction.However, the most common procedure for achieving phase matching is to make use of the birefringence displayed by many crystals.Definition: Birefringence is the dependence of the refractive index on the direction of polarization of the optical radiation.Here we only consider the case of uniaxial crystals. Ordinary light Extraordinary light (5) Index ellipsoidNegative uniaxial crystalPositive uniaxial crystalBut there are two choices for the polarizations of the lower-frequency waves. One choice is the type I phase matching for which the two lower-frequency waves have the same polarization; Another one is the type II phase matching, where the two polarizations are orthogonal. The possibilities are summarized in Table 2.3.2. As an illustration of angle phase matching, we consider the case of type I second-harmonic generation in a negative uniaxial crystal, as shown in Fig. 2.3.3.The phase matching condition (4) then becomes(6) or(7) This equation shows how the crystal should be oriented in order to achieve the phase-matching condition.With a little bit algebra, we obtain(8) As a result, ordinary and extraordinary rays with parallel propagation vectors quickly diverge from one another as they propagate through the crystal. This walk-off effect limits the spatial overlap of the two waves and decreases the efficiency of any nonlinear mixing process involving such waves.2. Verify Eq. (1.5.30a) in Page 41(9) (10) (11) Substitution of Eq. (11) into Eq. (10), with some manipulation, yields(12) (13) By expanding the matrix equation (12), we have (14) Considering that(15) we finally obtain (16) which is just the same formula as Eq. (1.5.30a) in page 41.逆水行舟用力撑逆水行舟用力撑, ,一篙松劲退千寻一篙松劲退千寻; ;古云此日足可惜古云此日足可惜, ,吾辈更应惜秒阴。吾辈更应惜秒阴。 董比武董比武 Exercises(作业):PleaseverifyEq.(1.5.30b)inpage41.