Читать книгу Liquid Crystal Displays - Ernst Lueder - Страница 23

A planar and harmonic wave of light (Lueder, 1998a; Born and Wolf, 1980; Yeh, 1988) is assumed to propagate through anisotropic liquid crystal layers with refractive indices n_|| and n┴. Together with a polarizer and an analyser, they generate grey shades controlled by a voltage. For the derivation of these electro-optic effects, we first need to consider the basic equations for planar harmonic waves. A harmonic electric field E(t) with amplitude E₀ is given by the scalar

(3.1)

where t is the time, ω the angular frequency and φ a phase angle. A travelling planar wave is a harmonic oscillation in a plane A in Figure 3.1 in the distance propagating in the direction of the unit vector with the speed is the vector from the origin to any location in the plane. Hence, replacing t by yields the equation for the planar harmonic wave in space and time as

(3.2)

Maxwell’s equations provide for the speed of light in vacuum

(3.3)

Figure 3.1 The plane A in which a planar wave travels with speed c₁ and wave vector parallel to the normal

and in materials

(3.4)

where μ₀, ε₀ are the permeability and permittivity in vacuum, and μ_r and ε_r are the pertinent relative constants in materials. Further, for μ_r = 1 the refractive index is , as known from Equation (2.6).

The wavelength λ in vacuum is given by

(3.5)

where f= ω/2π is the frequency. Equations (3.4), (2.6) and (3.5) provide for μ_r =1:

(3.6)

which results in the wave vector from Equation (3.2) with Equation (3.6) as

(3.7)

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 to and , respectively.

The two components can be represented by a column vector, which is called the Jones vector (Jones, 1941):

(3.23)

In a physical sense, J is not a vector as so far it consists only of two components and the vector product does not apply. It can also be termed a Jones matrix. According to Equation (3.10), the scalars associated with the Jones vector are

(3.24)

and

(3.25)

In later investigations, the rectangular coordinates ξ and η will have to be rotated by an angle a into the new rectangular coordinates x and y, as shown in Figure 3.3. The vector E with the components E_ξ and Eη is transformed into the components Ex and Ey in the x-y-plane according to

(3.26)

(3.27)

if rotation by a is positive in the counter-clockwise direction. The roman numerals in Equations (3.26) and (3.27) indicate the sections marked in Figure 3.3, thus providing Equations (3.26) and (3.27). The matrix equation for (3.26) and (3.27) is

(3.28)

R(α) is the rotation matrix, and R(−α) stands for a rotation in clockwise direction.

Figure 3.3 Rotation of the ξ−η coordinates by a into the x-y coordinates