The intensity fringes of three coupled waves in crystals

Two different kinds of interaction between three waves D0, Dsub(h) and Dsub(g) in a perfect crystal are investigated in the case of Laue scattering using the Takagi-Taupin equations. Polarization effects (coupling between sigma and π waves) are neglected, and it is assumed that incoming vacuum wave D0sup((e)) has a small wave-front area whose spatial extension is simulated by a point source on the crystal surface. The solutions of the diffraction equations thus constitute the boundary-value Green functions for the wave fields. In the first case it is assumed that Dsub(g) is only indirectly coupled to D0. In the second case energy is allowed to be exchanged between D0 and Dsub(h) and between D0 and Dsub(g), but no Dsub(h) - Dsub(g) interaction is present. In both of these situations the field amplitudes are given by expressions that contain simple products of zeroth- and first-order Bessel functions. It is suggested that the intensity pattern can be observed directly. The transition to an incoming plane wave is outlined, and it is also demonstrated that the hyperbolic intensity fringes generated by two spherical waves can be deduced from the derived expressions. (orig.)