Increased energy costs in combination with limitedpower capacity and the obligation to honor the KyotoProtocol has led to increased interest in energy-efficient building technologies. An important contri-bution to a building’s annual energy balance isprovided by the Sun. Solar irradiation can contributeto heating in the winter but unfortunately also tooverheating in the summer. The use of modern solarcontrol glazing with low-E coatings can considerablyreduce energy consumption by allowing the solar ra-diation to enter through windows in the winter or byreflecting most of it back to the outside in the sum-mer. Control of glare and daylighting inside a build-ing are also important issues. It is well known thatnatural daylight is important for people who spendtheir time indoors. It is a difficult problem to providenatural daylight without creating problems with
glare at the same time. Therefore daylight should beprovided by diffuse light, and direct radiation shouldbe avoided. Diffuse light is more relaxing and is alsomore evenly distributed to different parts of the in-terior. Diffusing glass and plastic sheets have beenavailable for a long time, but a general problem isenhanced backscattering, which seriously reducesthe transmittance of such devices.
This paper is a continuation of our previous studyof transparent refractive-index matched micro(TRIMM) particles as the scattering element intranslucent sheets for use in lighting applications1
and demonstrates and validates a ray-tracingmethod for simulation of the scattering profile at nor-mal angles of incidence. The initial study was limitedto normal-incidence spectral data, and in this paperwe examine the actual scattering profile and the ef-fects of varying the angle of incidence for both hemi-spheric and directional transmittance. A number ofapplications in lighting technology follow from theability of these sheets and polymers to scatter almostexclusively in the forward direction. Translucentsheets based on nonabsorbing polymer microparticlesrandomly dispersed in a transparent polymer haveseveral advantages over traditional pigmented trans-lucent sheets. Low backscattering and negligible ab-sorption are the main advantages achieved byreplacing traditional white pigments with TRIMMparticles. Not only do these properties yield high
J. C. Jonsson ([email protected]) and A. Roos arewith the Department of Engineering Sciences, Uppsala University,P.O. Box 534, SE-751 21 Uppsala, Sweden. G. B. Smith and C.Deller are with the Department of Applied Physics, University ofTechnology, P.O. Box 123, Broadway, Sydney, New South Wales2007, Australia.
Received 10 March 2004; revised manuscript received 25 No-vember 2004; accepted 9 December 2004.
transmittance but they also allow for easy tailoring ofthe output-light distribution without adversely af-fecting the number of lumens transmitted.
The scattering profile in general depends on sheetthickness, particle size, particle concentration, andparticle refractive index. In this case particle size is ofmuch less importance than for traditional scatterers,and in large enough particles (greater than approxi-mately 6 �m) it can be neglected. It would be prefer-able to have a closed analytical expression for thescattering profiles that contain the material param-eters of interest but, because of multiple scattering,finding such an expression is not an easy task, if it ispossible at all. Instead it is necessary to resort to useof ray-tracing simulation programs for determinationof scattering profiles of materials still in the designphase. This approach is clearly preferable to actuallyproducing several components with a wide range ofconcentrations for experimental characterization.
The TRIMM particle sheets did not show any ab-sorption at normal angles of incidence, but it is ofinterest to verify that such would also be true foroblique angles of incidence. Loss of transmittanceowing to increased specular reflectance from thesmooth sample surfaces is expected. Side loss, i.e.,light escaping from a sample’s sides, can also be ex-pected to increase with increasing angle of incidence.
2. Theory
A. Integrating Spheres
Single-beam integrating spheres, such as thoseshown in Fig. 1, do not give correct reflectance andtransmittance values by simply dividing the samplesignal by the reference signal. Corrections used inthis paper were derived and are thoroughly describedelsewhere,2,3 but a short summary is appropriate.
The measured hemispheric transmittance Tm
(sample-to-reference signal ratio) through a singleport tends to be too high for several reasons. The mostobvious is that the sample itself reflects light thatwould otherwise escape from the sphere through theentrance port, which does not happen during the ref-
erence measurement. Corrections are also made forthe fact that the diffuse part of the transmitted lightis directly scattered by the sample. This is not thecase for the specular part and the reference measure-ment, for which the first sphere wall reflection isneeded to diffuse the light such that it enters thedetector’s field of view. Another correction factor isincluded to compensate for any non-Lambertian scat-tering property of the sphere wall. Although it is ingeneral true that a sphere wall is not a perfect Lam-bertian scatterer, this effect is small and x in Eq. (1)below can be assumed to be equal to 1. The finalexpression for the measured transmittance can thenbe expressed as
where Ss is the signal recorded during the samplemeasurement, Sr is the signal recorded from the ref-erence measurement, Rw is the reflectance of thesphere wall, fs is the fraction of the sphere area thatis occupied by the sample port, Rs is the sample hemi-spheric reflectance, x is the deviation from Lamber-tian behavior, and finally Td and Ts are the correctdiffuse and specular transmittance values, respec-tively.
By designating the hemispherical transmittanceTtot � Ts � Td � CTtot � �1 � C�Ttot, where C is thefraction of light that is specularly transmitted, onemay solve Eq. (1) for Ttot, which yields
The values of Rs and C in Eq. (2) must be measuredby other means or estimated by use of prior knowl-edge about the sample, whereas Rw and fs are prop-erties of the integrating sphere used.
The reflectance sphere used3 for the experimentsreported in this paper was designed with a center-mounted sample holder that can be rotated about itsown axis. This design, as for the single-beam trans-mittance sphere, also results in an overestimation ofthe hemispheric reflectance unless proper correctionsare applied. Correction is needed for the differenttreatments required for specular and diffuse reflec-tance as well as for port area, sample area, and angleof incidence. The published equation3 is applicableonly for opaque samples, whereas our samples hereare transparent and need an extra term. With anapproximate correction for the light that is twicetransmitted through the sample and reflected fromthe sample holder, the full expression for the totalreflectance, Rtot, becomes
Fig. 1. Top-view schematic of the integrating spheres used forhemispheric measurements of scattering samples at oblique anglesof incidence. (a) In the transmittance sphere, both sample andsphere rotate about an axis through the entrance port. (b) For thereflectance sphere, the sample (lighter shading) and the sampleholder (darker shading) rotate, but the sphere is fixed. Both figuresshow the angle of incidence, �, which is measured with respect tothe surface normal.
where f is the fraction of the area of the sample holderthat is not covered by the sample, A2 is the ratio of thesample holder’s area to the total sphere area, � is theangle of incidence, Rh is the reflectance of the sampleholder, and all other symbols are as in Eq. (1). It isimportant to take into consideration that both Ttotand Rh vary significantly with angle of incidence. Theterms A2, Rw, and Rh are sphere parameters that areall sample independent. Equation (3) lacks any cor-rection for side loss, which is recorded in the Ss termbecause light that exits through the edges is con-tained inside the sphere and is hence detected.
As Ttot and Rtot depend on each other, it is notobvious how best to find the correct values. A sug-gested strategy is to measure diffuse and total trans-mittance and reflectance for normal incidence with adouble-beam integrating sphere, which is a standardaccessory that is available for most commercial spec-trophotometers. This gives information about the re-lation between scattered and diffuse light (the Cfactor) as well as a value of Rtot that one can use tofind Ttot with Eq. (2). Corrected Rtot values can then befound by use of Eq. (3) for each angle of incidence.
B. Goniometer Scatterometer
Hemispheric transmittance and reflectance are onlypartly of interest for the performance of a translucentsheet. The scattering angle profile that shows howdiffuse the transmitted light becomes is not deter-mined by use of an integrating sphere. Goniometerscatterometers have to be used to measure the angle-resolved scattering intensity and to determine thescattering profile.
The TRIMM sheets studied here are homogeneousand isotropically scattering; hence it is sufficient tomeasure only polar angle � from zero to 90° with afixed angle � for the normal angle of incidence. Foroblique angles of incidence the scattering profile be-comes asymmetric, and a larger part of the hemi-sphere must be measured. The scattering angle isalways calculated from the normal of the sample’ssurface as defined in Fig. 2.
C. Ray-Tracing Scattering Model
Ray-tracing simulations have been used to model thescattering behavior of the TRIMM-particle-dopedsamples, and the algorithm used to produce a scat-tering curve is shown in Fig. 3. Mie scattering theorycan be used to study a single scattering event, butthis does not suffice to describe properly the multiplescattering that occurs in the sheets studied. The sim-ulation was designed such that its results could becompared to such intensity profiles as those recordedwith angle-resolved scatterometers, i.e., the intensityfor fixed � versus scattering angle �, even though thesimulation is carried out in three dimensions andthus allows for rays to emerge at any �.
D. Angle-Resolved Light Scattering in Vector Space
Sorensen and co-workers have demonstrated4–6 thebenefits of studying Mie scattering with respect to thescattering vector rather than the experimentallymore convenient scattering angle. The actual param-eter used is the dimensionless quantity qR, where Ris the particle radius and q � 2k sin���2� is the scat-tering wave vector for scattering angle � and lightwave vector k. This kind of analysis of the TRIMM
Fig. 2. Schematic of the goniometer setup used for angle-resolvedscattering measurements. Angle of incidence � is defined as theangle between the incident light and the sample normal (dashedline). Scattering angle � is also defined relative to the surfacenormal. The sample normal corresponds to � � 0°. Positive anglesare given by the clockwise direction; negative angles correspond toangles in the counterclockwise direction from the surface normal.
Rtot �Rw(Ss�Sr) � Rsp[1 � Rw(1 � A2 cos �) � Rh(1 � f)A2 cos �]
1 � Rsp fA2 cos �� Ttot
2Rh, (3)
Fig. 3. Flowchart describing the ray-tracing simulation.
particle sheets is interesting in a wider scope thanjust this specific sample because it probes the theoryof multiple Mie scattering.
The only arguments for using the scattering angleas a variable are that it is intuitive (which is whyparts of the data presented in this paper are shown insuch a way) and convenient from an experimentalpoint of view, whereas wave vector q combines thescattering angle and the inverse wavelength. The in-verse of q is the probe length of the scattering.4 Mul-tiplying q by scattering particle radius R, which isalso included in Mie theory, yields a dimensionlessparameter that incorporates all variables in Mie scat-tering except the relation between the scatterers andthe index of refraction of the medium.
Sorenson and Fischbach have shown that the Miescattering from a single particle can be divided intothree power-law regimes4:
I � (qR)0 qR � 1,
I � (qR)�2 1 � qR � �,
I � (qR)�2 � � qR,
where the phase factor is � � 2kR|m � 1|, k is thewave vector, R is the particle radius, and m is theratio between particle and matrix refractive indices.These regimes hold for a single Mie scattering event,but how this translates to multiple scattering has notbeen thoroughly investigated. Knowledge about thisbehavior is expected to result in simpler interpreta-tion, even of multiple scattering data. For scatteringfrom particles of different sizes it is plausible to ex-pect a smearing of crossover points, making the tran-sition between one regime to another less sharp.However, in large TRIMM particles the R depen-dence is weak, but we retain it to allow for somecontribution from the minority of particles smallerthan 6 �m.
3. Experiments and Simulations
A. Studied Materials
In this study the number of samples studied wasreduced from 12 in the previous paper1 to 4 in anattempt to make the results clearer, given the greatercomplexity of the scattering data. We concentratedour efforts on the poly(methyl methacrylate) (PMMA)7N matrix samples with 1011F Plexiglas TRIMMparticles as scatterers. The samples have been de-noted N73 for a 3% weight concentration of TRIMMparticles and N77 for 7.3% TRIMM particle concen-tration. Thicknesses of 1 and 3 mm were studied forboth particle concentrations, and the size of all sam-ples was 5 cm 5 cm.
The refractive index of the matrix (PMMA 7N) de-creased from 1.496 at 400 nm to 1.488 at 700 nm. TheTRIMM particles’ refractive index was determinedwith an Abbe refractometer to be 1.507 at 633 nm,
resulting in an m of 1.011. The dispersion in refrac-tive index of the TRIMM particle material is assumedto correspond to that of the matrix in such a way thatm is constant with respect to wavelength.
A micrograph of the near surface of the high-concentration sample (N77) is shown in Fig. 4. Thespheres near the surface are expected to be slightlydeformed as a result of shear forces applied duringinjection molding.
B. Hemispheric Reflectance and TransmittanceMeasurements versus Angle of Incidence
The transmittance and reflectance for each angle ofincidence was measured with s- and p-polarized lightand calculated according to Eqs. (2) and (3). All val-ues presented are the arithmetic averages of the s-and p-polarized results.
The entrance port of the transmittance sphere iscircular, with a diameter of 42 mm, and the light spotis focused to a diameter of less than 3 mm. Focusingof the incident light causes a slight variation in theangle of incidence of 1°. The large ratio betweenport diameter and light-spot diameter means thatalmost all side loss7 will occur through the edge andthat there will be none by side-shifted light exitingthe front surface. The spot size increases with angleof incidence, and this limits the measurements to anangle of incidence of 70°.
The reflectance measurements were carried outwith the same spot size as the transmittance mea-surements. The sample holder, which is mounted inthe center of the reflectance sphere, was covered withblack tape on the sample side. The sample was at-tached with 1 cm 5 cm strips of transparentdouble-sided tape at the upper and lower parts of theholder’s surface. This left a small space between thesample and the black tape of the holder. The blacktape was characterized for the relevant incidence an-gles for use in Eq. (3). The double-sided tape behaves
Fig. 4. Micrograph of the N77 sample, showing the smearing ofsurface particles and an indication of a size distribution among theTRIMM particles. The full scale is 100 �m (10 �m per minor tick).
differently from the black tape, mainly in that itsticks to the sample. This complication is ignoredbecause the double-sided tape is assumed to be farenough away from the light spot, an assumption thatdoes not change with increasing angle of incidence, asthe light spot’s shape is not elongated toward thedouble-sided tape.
C. Angle-Resolved Measurements
All angle-resolved measurements were carried outwith a goniometer built at the University of Technol-ogy, Sydney.8 Because of the instrument’s design themeasurements were restricted to angles � from �90°to 70° while angle � was kept fixed. As the samplesstudied were isotropically scattering, this restrictionposed no problem. A fixed He–Ne laser was used incombination with a polarizer as a light source for allmeasurements. Results presented for oblique anglesof incidence are arithmetic averages of s- andp-polarized measurements.
D. Ray-Tracing Simulations: Comments on TheirImplementation
We carried out ray-tracing calculations to simulatethe He–Ne laser goniometer measurements. Themost frequent computation task in the program is thescattering event, which is calculated several times foreach ray, and optimization of this part was crucial toobtaining the correct results. The core of such anoptimization would be to get the random scatteringangle with as few run-time calculations as possiblefrom a generated random number, because makingfull calculations of the complete Mie formalism wouldbe far too time consuming and lead to unreasonablesimulation times. The relative intensity for an anglefollow from the relative probability that the ray willbe scattered in that direction. Knowing this probabil-ity, it is possible to create a function that correlates alinear random number to an actual scattering angle.Such a probability function was reduced to a simplelookup vector containing almost 5 106 elements to
Fig. 5. (a), (b) Hemispheric reflectance (thinner curves) and transmittance (thicker curves) values for several angles of incidence of theN73 samples of thicknesses 1 and 3 mm, respectively. (c), (d) 1 � Ttot � Rtot, which corresponds to absorption and side loss of thetransmittance measurements.
maintain a reasonable resolution for high angle scat-tering. This strategy sped up calculations consider-ably, requiring only a single scaling multiplicationoperation of a random number from 0 to 1 to producea scattering angle. But, as the lookup vectors arerather memory consuming, it is not possible to use alarge number of such vectors simultaneously; hencewe are limited to a discretized scattering distribution.
Restricting the simulation to a single wavelengthsimplifies the scattering event not only because thewavelength is constant but also because one does nothave to consider dispersion. The particle size distri-bution for the samples investigated was handled byuse of an equal mixture of two discrete sizes, R� 7.5 and R � 10 �m. As the ray reaches a particle,the particle’s radius is randomized to one of the twodiscrete sizes, and the angle distribution for that sizeis used.
As the ray passes through the sample and is scat-tered, new � angles are generated by use of the lookupvector and new � angles are randomly generated
Fig. 6. (a), (b) Hemispheric reflectance (thinner curves) and transmittance (thicker curves) values for several angles of incidence of theN77 samples of thicknesses 1 and 3 mm, respectively. (c), (d) 1 � Ttot � Rtot, which corresponds to absorption and side loss of thetransmittance measurements.
Fig. 7. Hemispheric transmittance and reflectance versus angleof incidence for 500 nm. The solid curves indicate the calculatedreflectance for a clear PMMA sample to emphasize the increasedeffect of scattering with increasing angle of incidence. The largedifference between N73D3 and N77D1 indicates that side loss ishigher for thicker samples.
from 0 to 2�. The new angles (�� and � ) are addedto the ray’s orientation by use of spherical geometryaccording to
�new � arccos(cos � sin �� � sin � sin �� cos � ),(4)
new � arcsinsin � sin ��
sin �new. (5)
When the ray is refracted out through the exitsurface its � angle is recorded by binning with 0.5°intervals. Rays exiting the side of the sample or back-scattered toward the light source are not collected inthe forward-scattering bins. The � angle value isdropped owing to sample symmetry. As all � anglesare recorded, the simulated intensity will be higherthan the measured intensity because the goniometeruses only a single �. This increase is proportional to
a factor sin �, by which the simulation result must bedivided to be comparable with the experimental data.The proportionality constant is divided by itself whenthe intensity is normalized, so it does not need fur-ther attention.
4. Results
A. Hemispheric Reflectance and TransmittanceMeasurements versus Angle of Incidence
The hemispheric transmittance and reflectance mea-surements shown in Figs. 5(a), 5(b), 6(a), and 6(b) allfollow the expected increase in reflectance, hence de-crease in transmittance, as the angle of incidenceincreases. Figures 5(c), 5(d), 6(c), and 6(d) show theamount of light that is absorbed or undetected owingto side loss. The absorption of PMMA is quite low, butit may increase slightly with increased path length oftravel through the samples. However, almost the full
Fig. 8. Experimental results for goniometer measurements of the scattering profiles versus angle of incidence for (a) N73, 1 mm; (b) N73,3 mm; (c) N77, 1 mm; and (d) N77, 3 mm.
contribution to the term 1 � Rtot � Ttot is due to sideloss.1
The spectra in Fig. 5(c) show an almost constantlevel for all angles of incidence except for the highest,70°. This is an effect of the combination of lowTRIMM particle concentration and thin sheet thick-ness, which results in negligible side loss at low an-gles of incidence. Figures 5(d), 6(c), and 6(d) all showa continuous increase in the side-loss term with in-creasing angle of incidence, which is expected. A com-parison of Fig. 5(d) (thick sheet with a low TRIMMconcentration) and Fig. 6(c) (thin sheet with a highTRIMM concentration) indicates that thickness playsa larger role in the side loss than does TRIMM con-centration. Comparing the transmittance and reflec-tance with the angle of incidence for a fixedwavelength, as shown in Fig. 7, makes the impor-tance of sample thickness for side loss even moreobvious. This is certainly due to the fact that theincident light hits the sample farther away from theport, giving it a longer distance over which to scatteraway from the entrance port.
B. Angle-Resolved Measurements at Oblique Angles ofIncidence
Results for angle-resolved scattering at oblique an-gles of incidence are presented in Fig. 8. Normalangle-of-incidence results have been included forcomparison. These results confirm that the scatteringis symmetric at the normal angle of incidence, asexpected for an isotropically scattering sample.Asymmetric broadening of the scattering profile isseen as the angle of incidence increases. It is inter-esting to note the similarity between Figs. 8(b) and8(c), which again indicates that the number of scat-tering events through a combination of thickness andconcentration is what produces the scattering profile.
This outcome is consistent with all other results, butit was of interest to verify that the thickness wouldnot result in peak broadening as a result of a longerpath length inside the sample.
C. Ray-Tracing Simulations for Angle-ResolvedMeasurements at a Normal Angle of Incidence
A comparison of measured data and ray-tracing re-sults is shown in Fig. 9(a). The slope toward higherqR is close to the predicted power �4 behavior. Thephase factor, �, is 2.5 and 3.4 for 7.5- and 10-�m radii,respectively, indicating the other two power regimesto be insignificant. A more detailed look at the power-law crossover regions is given in Fig. 9(b). The mostnoteworthy detail is that the power-law crossoverpoint seems to occur at approximately the same rel-ative intensity for all combinations of sheet thicknessand TRIMM particle concentration, whereas the scat-tering profile is shifted as concentration and thick-ness increase toward higher qR without muchdistortion of its shape. Another, more obvious, char-acteristic is that the 3-mm N73 sample and the 1-mmN77 sample are extremely similar, indicating thatthe number of scattering events is critical for theprofile, regardless whether this number comes from ahigh concentration or an increased thickness.
5. Conclusions and Outlook
A thorough characterization of a translucent sheetwith TRIMM particles has been carried out for acombination of different concentrations and sheetthicknesses. Hemispheric and angle-resolved opticalproperties at oblique angles of incidence agree withearlier measurements at normal incidence, which im-plies that the material is well suited to use in lightingapplications.
The possibility of simulating the scattering profile
Fig. 9. (a) Comparison of ray-tracing simulation and measured scattering intensities for a normal angle of incidence for all four samples.(b) Crossover regions for three of the simulated scattering intensities versus qR. Linear fits for low and high qR regions are shown. Valuesfor the 1�mm N77 sample have been left out because of their similarity to those of the 3�mm N73 sample.
of a sheet of given thickness, particle concentration,and particle refractive index is useful for the design ofnew materials and structures by use of TRIMM par-ticles. The inverse situation in which material pa-rameters can be obtained from a desired scatteringprofile is still not possible and may not be achievableconsidering the complexity of the scattering in thesematerials.
All results, experimental and from simulations, in-dicate that the combination of sheet thickness andparticle concentration that produces the effectivenumber of scattering events is what controls the scat-tering profile. After this paper was submitted, wecompleted an investigation of how the scattering pro-file scales with this scattering number.9 Briefly, it canbe said that with a high enough scattering number,the shape will break down.
Simulation of the scattering profile for oblique an-gles of incidence can be handled by the ray-tracingsimulation program because both � and � for each rayare calculated, but the lack of symmetry for a non-normal angle of incidence makes it harder to drawgeneral conclusions.
This study was financed by the Swedish Founda-tion for Strategic Research through the researchschool Advanced Micro Engineering, The SwedishResearch Council for Environment, Agricultural Sci-ences and Spatial Planning, and an Australian Re-search Council Linkage grant.
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