ANALYTICAL HAZE SIMULATION OF OPTICAL FILMS WITH 2D
PARABOLOIDAL MICROLENS ARRAY
Jeng-Feng Lin, Chin-Chieh Kang, and Shih-Fu Tseng
Department of Electro-Optical Engineering, Southern Taiwan University, Tainan, Taiwan
E-mail:jengfeng@mail.stut.edu.tw, 97-EC-17-A-05-S1-114
ABSTRACT
This paper describes haze simulation of optical films with 2D paraboloidal microlens array by
analytical formulae and optical simulation software ASAP, respectively. Derivation of the
analytical formulae assumes every paraboloidal lens is independent, i.e., we ignore the situation
that outgoing rays from a paraboloidal microlens can hit nearby paraboloidal microlenses.
Comparison between formulae calculation and ASAP simulation shows that they are close.
INRODUCTION
Under the pressure of cost down, multifunctional optical films, which combine light diffusion
and luminance gain, have become key components in LCD backlight units. Most of
multifunctional optical films are composed of regularly or randomly distributed conic
microlenses [1, 2, 3, 4]. In this paper, first the behavior of normally incident rays inside the
optical film is examined. Then formulae for haze calculation and haze simulation by ASAP are
presented.
The schematic diagram of the optical film with 2D paraboloidal microlens array is shown in
Fig. 1. The substrate is PET with thickness of 180 mm and refractive index of 1.49. Above the
PET there are a thin layer with thickness of 20 mm and microlens array, and both are made of
cured UV resin with refractive index of 1.56.
microlens
cured UV
resin
20μm
180μm
Figure 1. Schematic diagram of the optical film with 2D paraboloidal microlens array.
RAY TRACING
For haze measurement light is essentially normally incident to the sample. Therefore, as
shown in Fig. 2, we assume the light is normally incident to the left half of the paraboloidal
microlens. Since the microlens is rotationally symmetric and the light is normally incident, we
only need to do the ray tracing on a plane which contains the optical axis (z axis).
Derivation of the analytical formulae assumes every paraboloidal lens is independent, i.e., we
ignore the situation that outgoing rays from a paraboloidal lens can hit nearby paraboloidal lenses.
We assume the light is normally incident to the left half of the paraboloidal microlens. As
mentioned above, reflection at the parabola has three cases. Therefore the derivation of the
analytical formulae also has three cases.
(a) k < 0.835
This is the case c for reflection at the parabola, i.e., the incident ray is refracted at A and totally
reflected at B. Through some mathematical manipulations and numerical approximations, the haze
can be expressed as
0.0144
,
Haze  (1 
)

ff
2
k
(1)
where k is shape factor and ff is fill factor defined as the percent of the substrate that is occupied by
the microlens. Apparently, haze is proportional to fill factor and increased with higher shape factors.
(b) 0.835 < k < 1.44
Basically, when 1.44r<x1<x1c, the incident ray is almost totally reflected to the substrate. Hence
incident ray in this region can not contribute any diffused light and haze is decreased with higher
shape factors. The haze can be expressed as
0.683
,
Haze 
0.697  (1 / ff  1)k 2
(2)
(c) k > 1.44
When -Rb<x1<1.44r, the power of the incident ray is almost refracted to the air. Hence incident
ray in this region contributes diffused light and haze is increased with higher shape factors. The
haze can be expressed as
k 2  1.391
,
Haze  2
(k / ff )  1.377
(3)
From the above derivation, we obtain two important observations: (a) shape factor is the structure
parameter that affects haze, not radius of curvature or Rb alone; (b) haze is always increased with
higher fill factors. Assume Rb = 25mm and ff = 81%. Haze versus various radius curvatures is
shown in Fig. 5. Clearly, to have a reasonable haze, k should be away from 1.44. However,
considering both haze and luminance gain, k>1.44 is required.
For Rb = 30mm and ff = 81%, haze is simulated by analytical formulae and optical simulation
software ASAP, respectively. Haze versus various radius curvatures is shown in Fig. 6. For
simulation by ASAP, the microlenses are randomly distributed by a dedicated algorithm [6] and the
situation that outgoing rays from a paraboloidal lens can hit nearby paraboloidal lenses is not
ignored. Figure 6 shows simulation results from analytical formulae and ASAP are close.
Figure 2. Ray tracing inside a paraboloidal microlens
We can express the parabola in Fig. 2 as z = x2/2r, where r is radius of curvature and is
negative. Let Rb denote the radius of the circular bottom of the microlens and define shape
factor k as k = Rb/|r|. As shown in Fig. 2, the incident ray is normally reflected back to the
substrate after consecutive reflections at points A and B. Therefore, reflection at A and B can
have three cases: (a) when x2c<x1<x1c, the incident ray is totally reflected at both A and B, (b)
when x1c<x1<0, the incident ray is refracted at A and totally reflected at B, (c) when x1<x2c, the
incident ray is totally reflected at A and refracted at B.
Figure 5. Haze simulation by analytical formulae Figure 6. Haze simulation by analytical formulae
(Rb = 25mm and ff = 81%).
and ASAP (Rb = 30mm and ff = 81%).
HAZE SIMULATION
CONCLUSIONS AND ACKNOWLEDGEMENT
Figure 3 shows a schematic of hazemeter employed in the ASTM D1003 [5]. In haze
simulation, as shown in Fig. 4, the integrating sphere and photodetector on it in the real
hazemeter are replaced by an absorbing sphere, which totally absorbs any ray hit on it. In
addition the incident beam is simplified as a collimated beam. According to ASTM D1003, the
exit port subtends an angle of 8° at the center of the entrance port. Further more, we assume no
light scattering due to the hazemeter when specimen of optical film is absent. Then haze is the
percent of transmitted light that is scattered outside the exit port.
We have derive formulae for haze simulation under the assumption that outgoing rays from a
paraboloidal microlens can hit nearby paraboloidal microlenses can be ignored. Comparison
with ASAP simulation shows that results from both simulation are close. In addition, formula
derivation indicates that shape factor is the structure parameter that affects haze, not radius of
curvature or Rb alone. To have a reasonable haze, the shape factor (k) should be away from 1.44.
However, considering both haze and luminance gain, k>1.44 is required. This research is
sponsored by the Ministry of Economic Affairs and the project ID is 97-EC-17-A-05-S1-114.
REFERENCES
Figure 3. Schematic diagram of hazemeter.
Figure 4. Schematic for simulation of
hazemeter.
[1] http://www.mntehch.co.kr
[2] http://www.brightviewtechnologies.com
[3] Shun-Ting Hsiao, Po-Hung Yao, and Chung-Hao Tien, “High gain diffuser film with surface relief of
2D paarboloidal lens array,” IDW Digest, pp619-633, 2007.
[4] Joo-Hyung Lee, Jun-Bo Yoon, Joon-Yong Choi, and Sang-Min Bae, “ A novel LCD backlight unit
using a light-guide plate with high fill-factor microlens array and a conical microlens array sheet,”
SID’07 Digest, pp465-468, 2007.
[5] ASTM Standard D 1003-07: Standard Test Method for Haze and Luminous Transmittance of
Transparent Plastics, 2007.
[6] Shih-Fu Tseng, Design and simulation of multi-functional optical films, Master Thesis, Southern
Taiwan University, 2010.