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A study on design of wideband half wave plate using two
biaxial plates
Dong Eon Lim, Byung June Mun, Seung Yeol Hur and Gi-Dong Lee*
Department of Electrode Engineering, Dong-A University,
2, Hadan-Dong, Saha-Ku, Busan, 604-714, Korea
[email protected]
In this paper, we describe the change of the light polarization which is produced by optically
phase retardation materials as a half wave plate which can show the wideband property in
oblique incident. The film consists of two biaxial plates. We have characterized a film as half
wave plate retarder in the wide spectral range from λ=400 to 700 nm. To optimize the
configuration, we are performed on a Poincare sphere using Stokes vector and Muller matrix
method..
1. Introduction
Nowadays, an optical phase retardation plate is
one of the many optical devices or materials that
change the state of polarization state of the light
propagating inside these types of plates. The half
wave plate used to rotate the polarization state of a
polarized light which has a phase retardation of Γ=π,
rotates polarization of a linearly polarized light by
 (Azimuth angle  ). In order to consider the
wideband half wave plate, we will propose an
optical configuration with two biaxial films. In
general, the dispersion of the refractive index of the
optical components along the wavelength is
dependent on the material property. The
polarization states of the three primary colors(R, G
and B) generally differ from each other after
passing through retardation films because of the
different material and wavelength dispersion
properties. Therefore the phase dispersion property
in the entire visible wavelength should be
considered [1-3].
2. Calculation
In general, the retardation plate known to consist
of birefringent material having an optic axis
direction. The direction of the axis of the retardation
plate is defined by inner structure [4]. For
retardation plate, the difference between ordinary
wave and extraordinary wave is defined as the
birefringence ∆n. Such a difference is caused by
retardation of the slow light in comparison to the
fast light and it is proportional to the distance
passing through the within the plate. ∆n is given by
following relation [6]:
(1)
n  n  n
e
o
Where ne, no are extraordinary and ordinary
refractive indices of the birefringent material. A half
wave plate has a relative phase retardation value π
radian or 180° caused by difference of o and e wave.
The phase retardation of a birefringence material is
given by following relation:

2nd

(2)
Where d is the thickness of the birefringence
material, and λ is the wavelength of the incident
light. Note that the phase retardation Γ is a measure
of the change in the phase retardation as a result of
the propagation. The change of the phase
retardation of the entire visible wavelength range
used the Stokes vector and Muller matrix on the
Poincare sphere. Typical biaxial retardation film is
characterized with optical properties such as
retardation value, three principal refractive indices
and visible wavelength. In order to design the
optical retardation film, a relationship between three
refractive indices, Nz factor is important property.
The principal measure of property of biaxial film is
given by following relationship [6]:
 ( n  nz ) 
Nz   x

 (nx  n y ) 
(3)
Where nx and ny are in-plane refractive indexes. The
principal axes of the biaxial films are designed to
parallel or perpendicular to the axis of the polarizer
in the x-y plane, so as not to change the optical
property in the normal direction [5].
The one of the important issue is the dispersion of
the refractive index of the optical components along
the wavelength. In general, the dispersion is
dependent on the material property. The
polarization stats of the three primary colors(R, G
and B) usually differ from each other after passing
through the retardation plate. Therefore, to
minimize color shift at the oblique incidence, the
phase dispersion in the entire visible wavelength
should be eliminated [7].
biaxial films of NZ=3.8 and NZ=1.69. The viewing
angle is  =45,  =70. The optical axis of the lower
biaxial film is aligned with the transmission axis of
the incident polarizer and the optical axis of the
upper biaxial film is aligned with perpendicular to
the transmission axis of the analyzer [6].
Figure 3. Polarization state of the light passing through
the polarizer and polzrizer optic axis movement with
viewing angle
Figure 1. Optical configuration of the parallel polarizer
and biaxial films.
Figure 3 represent the polarization states of the light
passing through the incident polarizer on the
Poincare sphere. The start position K, when light
obliquely pass through the incident polarizer and H
is a analyzer transmission axis point. The key thing
the polarization axis of the polarizer and optical axis
of each optical film has the deviation angle δ from
the normal direction. In Figure 3, δ represents the
deviation angle of the polarizer. Thus the effective
principle axis of the optical components deviates
from the principle axis in the normal incidence by
angle δ. In terms of the polarizers, if we very small
birefringence approximation (ne≈ no), the deviation
angle δ in terms of  and o can be described as
below [11]:
3. Conclusion
Figure 2 shows the optical configuration of the
wideband half-wave film. It is consist of two biaxial
films between two parallel polarizers to examine
that two biaxial films achieve an excellent
wideband half wave plate in the oblique incident
light. The polarizer is used to choose a linearly
polarized light at a specific direction (  =45,  =70)
from the unpolarized light. In this part, the
polarizers are used to examine property of the half
wave plate.

sin 2 sin 2 ( / 2) 
2
2 
 1  sin  sin  
  arcsin 
(4)
Where  is the azimuth angle of the polarizer axis
of the polarizer, and o is the polar angle of the
incident light in the biaxial layer. ne and no represent
the extraordinary and the ordinary refractive index
of the polarizer and retardation film. From Eq. (3),
the deviation angle δ is maximized in the diagonal
direction (  =45˚). And effective angle of the
optical axis of the retarder in the oblique incidence
is changed as a function of the observation angle  .
According to the configuration shown in Figure 1,
we can get deviation δ from oblique viewing angle
[7].
Figure 2. Optical configuration of the proposed wideband
half wave plate
If we designed effective optical configuration, there
is no light leakage in the specific viewing angle.
The retardance of the proposed optical
configuration was determined from change of the
polarization state of the passing through the biaxial
films. Proposed configuration is consists of two
2
Table 1 is calculated optimized dispersion
properties of the biaxial films. In order to
accomplish the above calculation, the biaxial
retardation films satisfy nx>ny>nz, Rin=(nx-ny)*d,
where nx and ny are in-plane refractive indexes, Rin is
a in-plane retardation and d is a thickness. We
calculated the polarization position of the light as a
function of parameter Nz =(nx-nz)/(nx-ny) after the
right passes through the lower λ/2 biaxial and upper
λ/2 biaxial. The biaxial films have a normal
dispersion property.
Figure 4. Optical configuration and polarization path on
the Poincare sphere of the proposed wideband half wave
plate: (a) after passing through lower biaxial films (b)
after passing through upper biaxial films.
Figure 4 is explaining the polarization state and
polarization path on the Poincare sphere of the
proposed wideband half wave plate after passing
through two biaxial films. The symbols that blue
color represents 450nm with green color (550nm),
red color (630nm) are separately express visible
wavelength. The starting position is at position K,
where incident light pass through linear polarizer.
Then, after passing through first biaxial film, the
polarization states of the light move to positions Br,
Bg and Bb. The polarization position B move to
position Cr, Cg and Cb after passing through upper
λ/2 biaxial film. The final positions of the
polarization state of the light passing through two
biaxial films in wavelength R, G, B should be
coincident to the same position H. Therefore, we
can put the polarization positions of the light in
entire wavelength range by controlling the
retardation vale of the two biaxial films. The
process for the optimization on the Poincare sphere
is depicted in Figure 4. This show the polarization
states of the light passing through the two biaxial
films which is the desired final destination of the
perfect wideband half wave plate [8].
Figure 5. Polarization distribution of the light: (a) after
passing through the lower λ/2 biaxial film (b) after
passing through the upper λ/2 biaxial film as a function
of Nz factor
Figure 6. The variation of the calculated phase
retardation as a function of the wavelength.
When the Nz factor of the lower λ/2 biaxial film
change from 3.5 to 4.5, and the Nz factor of the
upper λ/2 biaxial film change from 1 to 2. Here, the
Nz factor should be considered to ensure that
polarization move to circle S2 ~ S1 . From the
calculated result, we confirmed that the first
condition for compensating for oblique incidence
can be satisfied with lower λ/2 biaxial film with
Nz=3.8 and upper λ/2 biaxial film with Nz =1.69 in
the Figure. 5 [9].
Table 1. Calculated optimized dispersion properties of
the optical anisotropy of the biaxial films
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Figure 7. The variation of the calculated transmittance as
a function of wavelength for wideband half wave plate.
Figure 6 and Figure 7 show the calculated phase
retardation and optimized wideband half wave films
used for visible wavelength. In the Figure 6, two
biaxial films have a normal wavelength dispersion
property. These two layer structure of biaxial film
has a relative phase retardation value π radian or
180°. To measure the property of the wideband half
wave plate, we designed two biaxial films between
two parallel polarizers. In the Figure 7, we proved
that the proposed wideband half wave plate has a
perfect wideband half wave plate because the light
passing through the two biaxial films eliminated by
analyzer’s absorption axis [10].
4. Conclusion
In this study, we presented several proposed
designs for wideband half wave plate using two λ/2
biaxial films. In this study, we can be minimized the
color shift at the entire visible wavelength for λ/2
plate by using two biaxial plate. To measure of the
phase retardation, we design two biaxial films in the
two parallel polarizers. We proved the
compensation films used for phase dispersion of the
entire visible wavelength range using the Stokes
vector and the Muller matrix on the Poincare sphere
[3].
5. Acknowledgements
This work was supported by the Samsung
Electronics Company.
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