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      with

      (3.8)

      and for vacuum

      (3.9)

The wave vector parallel to indicates the direction of the propagating wave. Its insertion in Equation (3.2) provides

(3.10)

      or

(3.11)

      and with

(3.12)

(3.13)

Equation (3.11) reveals that is the phase angle at distance r. The locus of constant phase Φ is determined by ., or as φ is a constant, by

      (3.14)

According to Figure 3.1, this is a plane perpendicular to , which is called the surface of constant phase, or the wave surface for short. The vector lies in this plane, as assumed after Equation (3.1). Hence, is perpendicular to . A more analytical statement follows after Equation (6.12).

In further calculations dealing with the propagation of the wave for varying r, the quantities A₀ and A₁ in Equation (3.13) can be omitted, as they do not change with r and have to be added again in the result, as given by Equation (3.13). Therefore, it is sufficient to deal only with the complex phasor

(3.15)

      which contains the amplitude E₀ and the change of phase in the distance r. Those phasors are widely used in optics (Born and Wolf, 1980), and in electric circuits fed with sinusoidal voltages in electrical engineering.

In the ξ-η-plane in Figure 3.2, the vector E of the electrical field is represented by the complex phasor P₀, with the complex components Pξ and Pη as

(3.16)

      where ξ₀ and η₀ are the unit vectors in the axes of the coordinates. The components are

(3.17)

      and

(3.18)

Figure 3.2 The phasor P₀ representing the vector E of an electrical field

In an anisotropic medium, the wave vector in Equation (3.7) is different in the ξ- and η-direction, with the unit vectors and as the refraction indices nξ and nη differ. Hence, we obtain from Equation (3.7) the wave vectors

      (3.19)

      and

      (3.20)

yielding with Equations (3.15), (3.16), (3.17) and (3.18)

      (3.21)

      and

      (3.22)

      Note that and are perpendicular

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