Page 670 - Introduction to Information Optics

654 11. Information Display with Optics

uniaxial crystal with n e and n 0 representing the refractive indices of the
extraordinary axis and the ordinary axis of the crystal. The angle between the
extraordinary axis and the x-axis is called the twist angle; assume that the twist
angle varies linearly with the rotation of z. The rotation of each layer then
becomes A$ = aAz, where a is the twist coefficient (degree per unit length). Let
us say the mth layer that is located at z = z m = mAz, m = 1, 2,..., N is a wave
retarder the extraordinary axis of which is rotated by an angle 8 m — mA9 with
the reference x-axis. This mth layer is expressed by the successive multiplica-
tions of Jones matrices:

1.45)

where

pr( )
M (Az) -
~
0
represents the phase retardation of the mth layer, and k v is the wavenumber of
light in free space. The overall Jones matrix of the twisted nematic liquid
crystal is expressed by the successive multiplication of the mth layer's Jones
matrix:

Since M mt{-0 m)Mr<*(9m-i) = **«« + O m^) - M ro( (-A#), we get

T - M ro
exp(-/£ T l 0
exp( — ,/0 Az)

cos aAz — sin aAz
(11.47)
sin aAz cos aA

where 4> = (n 0 + n e)k v/2 and P = (n e — n 0)k v. Since a « ft as is the case in
practice, we can assume the rotation per Az is small enough for rotation