Ernst Lueder - Liquid Crystal Displays

We want to determine the lim for ε → 0 of this expression, providing an infinitesimally small thickness of the slices, and hence the exact solution. From Equation (4.14), we obtain

(4.15)

For an easier calculation of the lim for ε → 0, we transform

(4.16)

into a diagonal matrix

(4.17)

where D contains the eigenvalues, and the columns of M are the eigenvectors of T ( ε ) R ( ε ) (Specht, 2000). The eigenvalues ξ 1,2are obtained by | T ( ε ) R ( ε ) – ξI | = 0, where I the unity matrix, as

(4.18)

providing

(4.19)

Since | ξ 1| = | ξ 2| 1, the eigenvalues can be rewritten as

(4.20)

and

(4.21)

with

(4.22)

On the other hand, ξ 1and ξ 2in Equation (4.19)describe the transmission of a wave through a slice with the thickness dε in Equation (4.5). Therefore, with the wave vector of this slice, we obtain

(4.23)

and

(4.24)

with

(4.25)

This finally leads to

(4.26)

with k ′ in Equation (4.25)and dε in Equation (4.5).

The eigenvectors V ( V 1, V 2) with the components Vx 1,2and Vy 1,2are calculated from Equation (4.16)by solving

as

(4.27)

with the two arbitrary constants r 1and r 2. The transformation matrix M is, from Equation (4.27)

(4.28)

with the magnitudes of the eigenvectors || V 1|| and || V 2|| given by

and

or with ξ 1and ξ sin Equation (4.18),

(4.29)

and

(4.30)

with φ from Equation (4.22).

We continue with the normalized matrices M :

(4.31)

Based on Equation (4.17), we can represent T ( ε ) R ( ε ) with the known matrices D in Equation (4.26)and M in Equation (4.31)as

(4.32)

[ T ( ε ) R ( ε )] 2πd/pεin Equation (4.14)assumes the form

(4.33)

The evaluation of D 2πd/pε, with dε in Equation (4.5)and D in Equation (4.26), provides

(4.34)

K ′ in Equation (4.25)leads to

As both the numerator and denominator tend to zero for ε → 0, the application of Hopital’s rule is needed, yielding

(4.35)

The repetition of Hopital’s rule provides

As the denominator of the last equation is identical with lim ε → 0 k ′/(2 π/p ) in Equation (4.35), we obtain

(4.36)

ensuring

(4.37)