-
atiit bal
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in Michor: m
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ion ofight. Wtion ofn givina metirradisimuldiffereimula.
2740.
1Aihas the role of providing the light passing through the
liq-uid crystal matrix. Edge-lit BLUs have the advantage ofusing
only one or a few light sourcesa cold cathode fluo-rccttpmV
imitawadeerom
Bptt
iesboth zero and finite absorption LGPs. Generally, it is
dif-ficult to know the exact value of the outcoupling coeffi-cient,
since there could be a significant difference between
Kusko et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A
2015escent lamp (CCFL) or a LED array, respectively. Basi-ally, the
light emitted by the CCFL or LED array isoupled into an LGP, where
the radiation is confined byotal internal reflection. Radiation can
be extracted out ofhe LGP in the upward direction by
light-scattererslaced on the bottom of the LGP. These scattering
ele-ents have various shapes, such as dots, microlenses,-grooves,
prisms, etc.Finding the optimal spatial distribution of the
scatter-
ng elements is very important for obtaining a uniform lu-inance.
Intuitively, the density of the scatterers should
ncrease for larger distances from the light sources, sincehe
amount of radiation remaining in the LGP diminishest larger
distances because of the interaction of the lightith the previous
scatterers. Some researchers have usedpower law function [1] or a
polynomial function [2] toescribe the distribution of the filling
factor of the scatter-rs. The scatterer distribution can also be
obtained by it-rative methods consisting of choosing an initial
configu-ation of scattering elements and optimizing this
forbtaining a uniform luminance by multiple simulations
oreasurements [36].The approach described in this paper is
deterministic.y considering that an effective parameter, the
outcou-
the designed scatterer and the real one. In order to over-come
this problem we propose a method to derive the realoutcoupling
coefficient from measurements. This methodhas the advantage of
reducing the number of simulationsand measurements for obtaining a
uniform luminance.In our study we consider scattering elements
with
shape and size that do not change with their position. Theonly
parameter that varies is the distance between twoadjacent
scatterers (and implicitly, the density of the scat-tering
elements). For reasons of simplicity we have con-sidered scattering
elements as V-grooves extending overthe entire width of the LGP.
These grooves can be fabri-cated by molding with a stamper obtained
by silicon mi-cromachining [3,7]. It is noteworthy that the results
of theformalism proposed in this article are independent of
theparticular shape and geometrical characteristics of
thescattering element. In our analysis we consider thatV-grooves
are oriented perpendicular to the light propaga-tion direction,
such that the spatial distribution of thescatterers is
one-dimensional.This work is structured in two parts, Sections 2
and 3.
Section 2, in turn, has two parts. In Subsection 2.A
theanalytical function of scatterer distribution is derived
as-suming the light guide has no absorption, followed by theMethod
of determindistribution in edge-l
an analytic
Mihai Kusko,* Cristia
National Institute for Research Development*Corresponding
aut
Received May 28, 20posted July 19, 2010 (Doc. ID
We present a method to find the optimum distributrendering a
uniform distribution of the outcoupled lthe light propagation in a
waveguide with a distributhe waveguide. We have found a
differential equatioa uniform irradiance along the LGP, and we
proposeficient of an individual scattering element from thethe
validity of this model by performing ray tracingtributed according
to the solution of the proposedagreement between the analytical
results and the scalculate the output power of a given embossed
LGP
OCIS codes: 120.2040, 080.3685, 220.4830, 220.2
. INTRODUCTIONmajor component of liquid-crystal-display
(LCD)-based
llumination systems is the backlight unit (BLU), whichling
coefficient k0defined as the ratio between the in-ensity of the
light scattered out of the LGP and the in-ensity of the incident
fieldcaptures the complex
mia
1084-7529/10/092015-6/$15.00 2on of light-scattereracklight
units usingapproach
ko, and Dana Cristea
rotechnologies, Bucharest, 023573, [email protected]
cepted July 5, 2010;); published August 18, 2010
scatterers in an edge-lit lightguide plate (LGP) fore propose a
simple mathematical model describingscattering elements located on
the lower surface ofg the distribution of scattering elements
leading tohod to determine the value of the outcoupling coef-ance
(or radiance) measurements. We have verifiedations on an LGP with
the scattering elements dis-ntial equation, and we have found a
quantitativeted ones. Also this model has been used to directly010
Optical Society of America
nteraction of the radiation with an individual scatteringlement,
we have found analytically the distribution ofcattering elements
for obtaining a uniform irradiance forethod for extracting the real
outcoupling value from therradiance measurements. In Subsection 2.B
a similarnalysis is done for the case when the LGP has a
nonzero
010 Optical Society of America
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2016 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Kusko et
al.bsorption coefficient. Section 3 presents the results
ofay-tracing simulations performed with the softwareEMAX [8] and a
comparison with theoretical results ob-ained in Section 2.
. MATHEMATICAL MODEL. LGP with Zero Intrinsic Absorptionhe
system we mathematically model is an LGP consist-ng of a
transparent lightguide that on the lower surfaceossesses an
embossed distribution of scattering ele-ents. Although in the
simulations performed for testinghe validity of the model we have
used a specific scatterer,his model is general and it can be used
for a scatteringlement with any geometric characteristics. We
considern array of scatterers (see Fig. 1), where the input
fieldntensity is I0. At the scatterer with the index i the inputeld
intensity is Ii1, and the scattered field intensity isIi1, where k
is the fraction of the input field scatteredut of the LGP, and the
remaining intensity inside theGP is 1kIi1. In this way, knowing the
configurationf the scattering elements and the outcoupling
coefficient, one can calculate directly the intensity of the
fieldmerging from the LGP as a function of the position of
thecatterers, as well as the irradiance and radiance quanti-ies,
respectively. Since we consider a one-dimensionalodel we will work
with a density (intensity per transver-al direction) denoted for
convenience by I and referencedurther in the work as the
intensity.Using this model one can also determine the
configura-
ion of the scattering elements necessary to obtain a re-uired
distribution of radiance (or luminance in photo-etric units) of the
upper surface of the embossed LGP.t the scattering element with
index i the irradiance Q isefined as the ratio between the
outcoupled intensity andhe distance between two adjacent scatterers
x and isxpressed as a finite difference equation:
Q = kIi1
x=
I
x. 1
f the outcoupling coefficient is small we can apply theontinuous
limit so that the finite difference equation be-omes a differential
one. The value of Q is proportional tohe intensity of radiation I
propagating in the LGP, whichepends on the distance x and an
x-dependent attenua-ion coefficient S=k /x, so that
Q = dI
dx= SxIx. 2
If S is constant (meaning that the scatterers arequally spaced)
then the field intensity inside the LGPx and the Q value will
decrease with x exponentially. InfiFig. 1. Schematic diagram of the
scattering process.rder to obtain a uniform Q over the entire
propagationistance, it is necessary that the coefficient S have a
de-endence with propagation distance. The variation with xf the
attenuation coefficient can be easily deduced fromhe above
expression, since Ix must depend linearly on xuch that dI /dx is
constant. If we consider the length Leeded for a complete radiation
coupling out of the LGP,he intensity function with respect to
distance is writtens
Ix = I0L x
L, Q =
I0
L, 3
ith S presenting the functional dependence
Sx =1
L x. 4
he spacing between two adjacent scatterers and itsariation with
distance is given by the relation
x = kL x. 5
One can see from Eq. (5) that there are two parameters,he length
L of the LGP and the outcoupling coefficient k,hat completely
determine the scatterer configurationecessary for a uniform
irradiance over the entire lengthf the LGP. For an outcoupling
coefficient k constant overhe propagation length, the spacing x
varies linearlyith x. One can note that the density of scatterers
in-reases with x and eventually tends to infinity when x ap-roaches
L. It is noteworthy that relation (5) is valid alsoor an
x-dependent outcoupling coefficient kx. The scat-erer position can
also be obtained as a function of N, de-oted as the order number of
the scatterer with respect tohe origin. This can be done using
relation (5) by consid-ring an iterative approach.Another problem
of interest is the behavior of the irra-iance distribution for the
case when the scatterer con-guration is realized according to
relation (5) assuming aertain k that is different from the real one
k0. The reasonor investigating this problem is to provide a method
toeasure the real outcoupling coefficient. Let us assumehat one
places the scatterers according to Eq. (5), whereis a given
outcoupling constant not necessarily equal tohe real outcoupling
constant k0. The differential equa-ion describing the propagation
in the LGP is
dI
dx=
1
L xI, = k0/k. 6
he solutions for the intensity in the LGP and for the ir-adiance
are given by the following expressions:
Ix = I01 xL
, 7
nd
Qx =I0
L 1 xL1
. 8
By examining Eq. (8) one can distinguish two classes ofrradiance
behavior as a function of the ratio =k0 /k. The
rst class of solutions occurs when kk0. In this case the
-
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Kusko et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A
2017atio is greater than 1 and the irradiance goes to zero athe end
of the LGP. When k is much smaller than the realutcoupling
coefficient k0 the irradiance decreasesmoothly over the entire LGP.
When k is half of k0 the ir-adiance decreases linearly with x, and
when k is close to0 the irradiance exhibits a sharp downturn at the
end ofhe LGP. The second class of solutions occurs when the ra-io
is less than 1. In this case, the irradiance exhibits anpturn
toward the end of the LGP. From a practical pointf view, the
irradiance behavior as a function of the ratioallows the accurate
determination of the real value of
he outcoupling coefficient k0.By measuring the irradiance of an
LGP embossed with
catterers possessing an unknown k0 and distributed ac-ording to
Eq. (5), where k has a predefined value, and bylotting logQ versus
log1x /L one obtains a straightine whose slope is k0 /k1. Knowing
the value k one canhen determine the real value of the outcoupling
coeffi-ient. It is noteworthy that since this procedure
involvesog-log graphs, one needs the relative values of the
irradi-nce for determining the outcoupling coefficient.
. LGP with Finite Intrinsic Absorptionfter setting the formalism
for the simpler case of zerobsorption LGP, we will extend this
analysis to take intoccount a nonzero absorption coefficient. In
this case, theoss of intensity per unit length is expressed as
dI
dx= SxIx, 9
here represents the internal absorption coefficient ofhe LGP
material. If we also consider that Q=I0 /Leff,here Leff is an
effective length LeffL, we solve thebove differential equation and
obtain that the intensityemaining in the LGP is written as
Ix = I01 + 1Leffexp x 1Leff . 10f vanishes, one can reduce the
above expression to Eq.3), finding the case of zero absorption by
considering that=Leff and keeping the first-order term of the
expxeries expansion. For the case of uniform irradiance theondition
Q=constant is imposed. In this case, the attenu-tion coefficient S
is expressed as
S =1
Leff + 1exp x 1
. 11
he value of Leff may be arbitrary, but it is preferable toe as
low as possible in order to obtain a maximum valueor the
irradiance. The minimum Leff value can be ob-ained by imposing the
condition that the intensity in theGP at the end be zero. This
leads to the expression
Leff =expL 1
. 12
similar analysis with the zero absorption case is per-ormed for
extracting the real outcoupling coefficient k0. If
e consider that the scatterers are generated assuming tn
estimated outcoupling coefficient k we obtain the dif-erential
equation that describes the intensity loss pernit length:
dI
dx= I
Leff + 1exp x 1
. 13
e finally obtain
I = I01 + Leff expxLeff
exp x. 14
he irradiance is expressed as
Q =I0
LeffLeff +
1
1
expx
Leff
1
. 15
f vanishes the above equation reduces to Eq. (8) if wexpand expx
and Leff in Taylor series and we keep therst-order term. Since the
ratio appears in the exponentf Eq. (15), as in the zero absorption
case, a similar analy-is regarding the variation of irradiance with
x can beone.Similarly, one can easily obtain 1 by plotting logQ
ersus log1+ Leff1 Leff1 expx and extractinghe real outcoupling
coefficient. In the Section 3 this pro-edure will be applied to the
results obtained from rayracing simulations.
. SIMULATIONS AND COMPARISON WITHHE THEORETICAL CALCULATIONSn
the following the comparison between the mathemati-al model and the
ray tracing simulations will be pre-ented. The simulations were
performed using the soft-are ZEMAX. The embossed LGP has the
followingeometrical characteristics: length 1000 mm, thicknessmm,
and width 0.1 mm. The light was emitted by a two-ngle source with
an angle spread in the horizontal planef 0 (collimated light) and
in the vertical plane y-z of 90ith a Lambert distribution (see Fig.
2, upper panel). Theower of the source is 1 W and the wavelength is
550 nm.
ig. 2. LGP structure considered in simulation. Upper panel:
he embossed LGP; lower panel: the scattering element.
-
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2018 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Kusko et
al.bove the upper surface of the LGP a detector was placedeasuring
the irradiance in W/cm2. Under the lower sur-ace of the LGP another
detector was placed. The inten-ity scattered out of the LGP is
given by summing the ir-adiances measured in the upper and lower
detectors.hese values are compared to the results obtained usinghe
analytical formulas derived in Section 2. For simplic-ty,
throughout this work radiometric units have beensed. The
simulations were done in non-sequential mode,ith 10 million rays,
taking into account the polarizationnd using the option split
rays.We have considered scatterers of triangular shape,hich can be
obtained if one uses a stamp tool that can beasily fabricated by
anisotropic etching of V-grooves in a100) silicon substrate
followed by replication. The scat-ering elements are identical over
the LGP length. TheGP and the scatterer element are shown in Fig.
2. It ismportant to note that in the case of this particular
scat-erer most of the rays leave the LGP in the upward direc-ion
with angles relatively close to the normal. An impor-ant part of
the radiation energy is deflected through theottom of the LGP at
angles centered at 70 to the nor-al. We have used this scatterer
geometry only for dem-nstration purposes. However, from our
previous studiese have found that the analytical approach for the
non-bsorption case can be applied to other types of scatterers9].It
is also important to point out that Leff cannot have
he ideal theoretical value as expressed in Eq. (12), sincehe
scatterer density increases to the LGP end such thathe scatterers
will overlap each other. Therefore, thealue of Leff should be
chosen in order to avoid the overlapf the scatterers at the end of
the LGP. We have chosenhe value of Leff to be 1200 mm in our
simulations. Theimulated systems were LGPs with 80% internal
trans-ission after 1000 mm, and with an embossed distribu-
Fig. 3. (Color online) logQ versus log1+ Leff1 Leff1ion of
scatterers arranged according to Eq. (11) for the kalues of 0.004,
0.0008, 0.0012, and 0.002. We have con-idered that the relation S=k
/x still holds becausexpx1, meaning that the intrinsic absorption
overhe distance between two consecutive scatterers can beeglected.
It should be emphasized that we consider thatdoes not vary with the
distance, which means that theeld inside the LGP does not change
its angular distribu-ion. Further simulations show the validity of
this as-umption.The logQ versus log1+ Leff1 Leff1 expx
raphs are shown in Fig. 3. They are indeed straightines,
confirming the validity of the mathematical model.ne can note from
Fig. 3(a) that toward the end of theGP, the log-log curve for
k=0.0004 departs from atraight line. The corresponding Fig. 5(a)
shows that inhis region the values of the absolute irradiance
aremall, approaching zero. Therefore, in this case, we dis-
for (a) k=0.0004, (b) k=0.0008, (c) k=0.0012, (d)
k=0.002.OLstsFig. 4. (Color online) Ratio k0 /k versus 1/k.
-
carded these points from our analysis, fitting with astraight
line only the first 3/4 of the total length of theLt1et
r0mlc
ance is multiplied by a correction factor that has the roleof
compensating for the Fresnel losses, representing 4%,tntm
gtiptL
Fig. 5. (Color online) Comparison between the irradiances
obtained from the ray tracing simulations (circles) and calculated
ones (solidcurves) using formula (9b). (a) k=0.0004, (b) k=0.0008,
(c) k=0.0012, (d) k=0.002.
Ftkc
Kusko et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A
2019GP. From the slopes of these graphs, one can calculatehe ratios
k0/k=, and by plotting these values versus/k, one determines the
real value of the outcoupling co-fficient; this is represented in
Fig. 4. For these structureshe value of k0 is approximately
0.0017.We have also plotted the irradiance for the structures
ealized with k values of 0.0004, 0.0008, 0.0012, and.002. The
results illustrated in Fig. 5 show good agree-ent between the ray
tracing simulations and the ana-
ytical calculations done using Eq. (15). This agreementonfirms
the validity of the model. The theoretical irradi-
ig. 6. (Color online) Irradiance for an LGP with finite
absorp-ion embossed with a distribution of scatterers generated for
a=0.0017 obtained from simulation (circles) and theory (solid
urve). chat occur when the light is coupled into the LGP and
theumerical losses that occur in the simulation. The correc-ion
factor is found to be 0.91, and it provides the bestatch between
the theoretical and numerical results.With this value of the
outcoupling coefficient, one can
enerate the array of scatterers that was used in the rayracing
simulations. The irradiance is shown in Fig. 6 ands indeed constant
with an average numerical value of ap-roximately 0.78 W/cm2, in
fairly good agreement withhe theory. The irradiance obtained for a
non-absorptionGP is shown in Fig. 7. The correction factor is now
0.93.
ig. 7. (Color online) Irradiance for an LGP with zero absorp-ion
embossed with a distribution of scatterers generated for a=0.0017
obtained from simulation (circles) and theory (solidFtk
urve).
-
4. CONCLUSIONSIn this article we have set up an analytical
framework fordetermining the spatial distribution of scattering
ele-ments placed on the lower surface of an LGP necessary toobtain
a uniform irradiance of the edge-lit backlight sys-tem. The main
assumption of this approach consists ofconsidering that the complex
interaction of the light con-fined in an LGP with a scattering
element can be solelydescribed by a single effective parameter,
namely, the out-coupling coefficient k, which is the ratio between
thepower scattered out from the LGP and the power of theincident
field interacting with the scattering element. Inthis way, by
direct computation, one can obtain the spa-tial distribution of the
light power emitted through theupper surface of the LGP.
Conversely, for small k, a dif-ferential equation whose solution
gives a constant radi-ance can be written for both zero and finite
intrinsic ab-sorption of the LGP material. The analysis of the
relativeradiance curves given by an LGP presenting a given
dis-tribution of scattering elements provides an accurate
de-termination of the outcoupling coefficient. This
analyticalmethod and its basic assumptions have been
quantita-tively validated by ray tracing simulations in which
wehave considered a one-dimensional system with V-grovesas
scattering elements and incident light collimated inthe horizontal
direction. While the one-dimensional casehas been chosen for
purposes of simplicity, this analyticalapproach can be extended to
scattering elements with afinite width, such as prisms or
microlenses. The methodpmi
ACKNOWLEDGMENTSThis work was supported by European
Project(FP7-NMP) FlexPAET, contract No. 214018/200. Wewould like to
thank Cristoph Baum and Kari Rinko fortheir valuable advice and
useful discussions.
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al.resented here has the advantage over the iterativeethods of
minimizing the number of the testing emboss-
ng.8. www.zemax.com.9. European Project (FP7-NMP) FlexPAET,
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