2 1
1 b
P
1cos
sin
a
dP
1cos
d 2
, (3)
a
m mm
m
mm
,
b
mm m
mm
m
, (4)
2rnm0, (5)
m nsnm, (6)
which are derived from Mie scattering theory,4,5
where I, m, and K, m are scattering intensity
and scattering efficiency, respectively, is a size parameter,
m is the relative refractive index between
particle ns and matrix nm, r is the particle radius,
and is the wavelength of incident light in the matrix.
P
lcos is a Legendre polynomial and and
are the first two orders of the Ricatti–Bessel functions.
According to Mie scattering theory,4,5 there
are two important properties of the scattering phenomenon
caused by the spherical heterogeneous
structure: 1 As size of the heterogeneous structure
becomes larger, the ratios of forward and backward
scattering intensity to total scattering intensity become
higher and lower, respectively; this effect was
remarkable when the size of the heterogeneous structure
was comparable with or greater than a wavelength.
2 Although the scattering efficiency was
described to be inversely proportional to the fourth
power of the wavelength based on Rayleigh scattering
theory,5 the actual scattering efficiency has a
more-complex relation to wavelength. We refer to
these properties as Mie scattering properties MSP
in this paper to distinguish them from those that can
be described based on Rayleigh scattering theory.
Fig. 1. Bulk HSOT polymer and conventional transparent bulk
polymers. a Light injected into the HSOT bulk polymer is multiply
scattered and homogenized because of the heterogeneous
structures of the HSOT and then emerges as a directive illuminating
light. b Injected light simply passes through conventional
transparent bulk polymer.
1 December 2001 Vol. 40, No. 34 APPLIED OPTICS 6275
Single-scattering profiles calculated based on Mie
scattering theory are shown in Fig. 2. For scattering
profiles in the forward direction 0°–90° and in the
backward direction 90°–180°, Mie scattering theory,
which is applicable to a spherical heterogeneous
structure with any size, gives asymmetric singlescattering
profiles. Although scattering has been
thought to distribute light in all directions, one can
achieve a forward-directional single-scattering pro-
file in which scattered light is concentrated in a forward
small angle by controlling the heterogeneous
structure. As a result, injected light can be guided,
multiply scattered while directivity is maintained,
and homogeneously radiated in a specific direction
from its surface.
Scattering-efficiency curves of the diameter of a
spherical heterogeneous structure at 615, 545, 435
nm, which correspond to red, green, and blue light,
are shown in Fig. 3. Generally, we tend to think,
based on Rayleigh scattering theory, that blue light is
always scattered more strongly than red light.
However, such is not always true and depends on the
size of the heterogeneous structure. By injecting
white light from a typical cold fluorescent lamp or a
white LED into the scattering medium that contains
the heterogeneous structure, curve A, we obtained
yellowish transmitted light with a lower color temperature
because blue light was scattered more
strongly than red light. This is the same phenomenon
as is responsible for the red sunset. However,
bluish transmitted light with a higher color temperature
was obtained in the scattering medium that
contained the heterogeneous structure, curve B, because
red light was scattered more strongly than blue
light. By controlling the diameter of the heterogeneous
structure to give almost the same scattering
efficiencies for red, green, and blue light, one can
obtain output light with almost the same color temperature
as that of injected light.
However, there were no materials and devices that
used the MSP effectively, because none provided the
precise control of the form and size of the heterogeneous
structure and of the distance of the heterogeneous
structures from one another that was
necessary. Scattered light from each spherical
structure interferes when the distance is small. Under
these circumstances the scattering form does not
follow Mie scattering theory. Thus we have proposed
using the HSOT polymer, which will enable us
to utilize the MSP. We formed the heterogeneous
structure in the HSOT polymer precisely by doping
transparent particles with a narrow distribution of
diameter sizes. When the distance among particles
is approximately or slightly larger than 100 times the
wavelength, single-scattering behavior of each particle
approximately follows Mie scattering theory.
Heterogeneous structures in conventional materials
such as polymer blends and copolymers had not been
optimized to utilize the MSP.
B. Multiple Scattering
We analyzed the light-homogenization effect caused
by multiple scattering in the HSOT polymer by using
the HSOT design simulation with the Monte Carlo
method6 based on Mie scattering theory.4,5 We
doped polymer particles to introduce the microscopic
heterogeneous structure. We used the Monte Carlo
method to analyze random and repeating processes
in the multiple-light-scattering phenomenon because
the Monte Carlo method is a powerful method with
which to solve problems that have no immediate
probabilistic interpretation.
By the Monte Carlo method we can define scattering
angle , expected photon path length L,
Fig. 2. Calculated single-scattering profiles based on Mie scattering
theory. The vector from the origin of the coordinates to each
curve is proportional to the logarithmic intensity scattered at the
corresponding angle. Size parameters, 1.7, 11.5, 69.2; relative
refractive index, m 0.965.
Fig. 3. Scattering efficiency curves of a single particle for 435-,
545-, and 615-nm wavelengths. Typical cold fluorescent lamps
have spectral peaks near these wavelengths. Relative refractive
index, m 0.965. A, B Particle diameters for heterogeneous
structures A and B, respectively.
6276 APPLIED OPTICS Vol. 40, No. 34 1 December 2001
probability-density distribution function F of scattering
angle, and extinction coefficient as
0