Crystals birefringent, propagation directions

In such birefringent crystals, the direction k of the wave propagation and the direction of the Poynting vector S = c q E x ), which is the direction of energy flow, do not coincide (Fig. 6.2). Only in the directions parallel or perpendicular to the optical axis the two vectors point into the same direction. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

In order to avoid laser waves propagating in both directions through the ring resonator, losses must be higher for one direction than for the other. This can be achieved with an optical diode [5.32]. This diode essentially consists of a birefringent crystal and a Faraday rotator (Fig. 5.18), which turns the bifringent rotation back to the input polarization for the wave incident in one direction but increases the rotation for the other direction. 

This condition can be fulfilled in unaxial birefringent crystals that have two different refractive indices no and n for the ordinary and the extraordinary waves. The ordinary wave is polarized in the x-y-plane perpendicular to the optical axis, while the extraordinary wave has its -vector in a plane defined by the optical axis and the incident beam. While the ordinary index no does not depend on the propagation direction, the extraordinary index n depends on the directions of both E and k. The refractive indices Uo, and their dependence on the propagation direction in uniaxial birefringent crystals can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric tensor. If these axes are aligned with the jc-, y-, z-axes, we obtain with n = the index ellipsoid. 

Fig. 5.101. (a) Index ellipsoid and refractive indices no and Hq for two directions of the electric vector of the wave in a plane perpendicular to the wave propagation k. (b) Dependence of Ho and He on the angle 0 between the wave vector k and the optical axis of a uniaxial positive birefringent crystal... 

Figure 6.1 a Index ellipsoid for uniaxial birefringent optical crystals, b Cutting through the ellipsoid in a plane that contains the optical axis and the propagation direction k... 

Figure 6.2 Directions of electric field E, polarization P, magnetic field B, wave propagation ft, and energy flow S in a birefringent crystal...

This condition can be fulfilled in birefringent crystals which have two different refractive indices n and n for the ordinary and the extraordinary waves. While the ordinary index ng does not depend on the propagation direction the extraordinary index n depends on both the directions of E and k. 

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