Chapter 3 Dichromatic BSDF
3.4 Summary
3.4 Summary
A simple but effective phosphor modeling of the pcLED is proposed. The major advantage of this study lies in that there is no need to formulate the complex physical mechanism of the phosphor scattering in a microscopic viewpoint. Instead, as long as the coated phosphor layer is available, the proposed methodology assisted by the measured BSDFs is able to characterize the phosphor properties with the direction and wavelength variables. By the Monte Carlo simulation, pcLED luminous intensity distribution and its angular CCT distribution can be predicted with high accuracy.
Closed agreement with a commercially available pcLED validates the proposed scheme, which certainly has impact for the LED development in illumination applications.
3.5 References
[1] S. Nakamura and G. Fasol. The Blue Laser Diode: GaN Based Light Emitters and Lasers, (Springer-Verlag, New York, 1997).
[2] R. Mueller-Mach, G. O. Mueller, M. R. Krames, and T. Trottier, “High-power phosphor-converted light-emitting diodes based on III-nitrides” IEEE J. Sel. Top.
Quantum Electron. 8, 339-345 (2002).
[3] http://www.philipslumileds.com/technology/whitelighting.cfm
[4] N. Narendran, Y. Gu, J. P. Freyssinier-Nova, and Y. Zhu, “Extracting phosphor-scattered photons to improve white LEDs efficiency,” Phys. Status Solid A 202, R60-R62 (2005).
[5] H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor based white-light-emitting diodes,” Appl.
Phys. Lett. 86, 243505 (2005).
[6] Y. Ito, T. Tsukahara, S. Masuda, T. Yoshida, N. Nada, T. Igarashi, T. Kusunoki, and J. Ohsako, “Optical design of phosphor sheet structure in LED backlight system,” SID Int. Symp. Digest Tech. Papers 39, 866-869, (2008).
[7] C.-H. Tien, C.-H. Hung, B.-W. Xiao, H.-T. Huang, Y.-P. Huang, and C.-C. Tsai,
“Planar lighting by blue LEDs array with remote phosphor,” Proc. SPIE 7617, 761707 (2010)
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[8] S. C. Allen and A. J. Steckl, “ELiXIR—Solid-state luminaire with enhanced light extraction by internal reflection,” J. Disp. Technol. 3, 155-159 (2007).
[9] S. C. Allen and A. J. Steckl, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
[10] K. Yamada, Y. Imai, and K. Ishi, “Optical simulation of light source devices composed of blue LEDs and YAG phosphor,” J. Light & Vis. Env. 27, 70-74 (2003).
[11] Y. Zhu, N. Narendran, and Y. Gu. “Investigation of the optical properties of YAG:
Ce phosphor,” Proc. SPIE 6337 (2006).
[12] J. de Boer, “Modelling indoor illumination by complex fenestration systems based on bidirectional photometric data,” Energy and Buildings 38, 849–868 (2006).
[13] C.-H. Tien, and C.-H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17, 723-732 (2009).
[14] Francois X. Sillion, and Claude Puech, Radiosity and Global Illumination, (Morgan Kaufmann Publishers Inc., San Francisco, 1994).
[15] H.-T. Huang, C.-C. Tsai, Y.-P. Huang, J. Chen, J. Lin, and W.-C. Chang,
“Phosphor conformal coating by a novel spray method for white light-emitting diodes as applied to liquid-crystal backlight module,” in proc. International Display Research Conference (Rome, Italy, 2009) 17.5.
[16] M. Shaw and T. Goodman, “Array-based goniospectroradiometer for measurement of spectral radiant intensity and spectral total flux of light sources,”
Applied Optics 47, 2637-2647 (2008).
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Chapter 4
Calculation Methodology for Energy Balance Equation
Although the optical scatter properties can be characterized by BSDFs, the integration of energy balance equation (Eq. 3.14), is a subject to discuss in global illumination. In this chapter, we propose a simple but effective methodology to implement the energy integration by using discrete summation. The main purpose is to get the suitable sampling number of a considered object in the relation of Fig. 1-1.
The discrete sampling grid was tested by a reference Lambertain source, which is available for practical experiment. We introduced the correlation coefficient to evaluate the predicted luminance distribution with the experimental measurement, and get a feedback to iterate the sampling number. The iterative algorithm was thus proposed to get the acceptable sampling grid. Here a commercial diffusing sheet of a LCD backlighting module was used to demonstrate the proposed algorithm.
4.1 Diffuse Scheme
Diffuse illuminations have been widely used for many fields, including projectors, liquid crystal displays, traffic signs, and luminaries. With the increasing demand of the illumination systems, the requirements for diffusing function have become more diverse, where the luminous flux is reformed into a defined radiance angles for the purpose of beam shaping, brightness homogenizing, antiglare, directionality adjustment, and so on. Meanwhile, the rapid progress in the manufacture technologies also led to many new diffusing components for various purposes. To characterize the
49
diffuse properties become essential for illumination design practice. However, a general and effective methodology for diffusing behavior in different area is sill insufficient.
The diffuse phenomena are mainly caused by the optical scattering. As Chapter 2 mentioned, scattering from the sheet components is caused by four mechanisms:
surface topography, surface contamination, bulk index fluctuation and bulk particulates. The physical phenomena involved in the scattering are most properly described by the Maxwell’s equations with appropriate boundary conditions. The modeling approaches can be conducted by either analytical or numerical solution. The analytical approaches include Kircihhoff approximation [1-4], the small perturbation method [1,3], the integral equation method [4], the small slope approximation [5,6], the facet method [7-9], and so on. Besides, the numerical methods, such as finite-difference time-domain (FDTD) [10,11] and Monte Carlo ray-tracing [12,13], have also been developed. However, only one kind of mechanisms, such as surface topography, can be considered in one physics model, where a few conditions are assumed before the calculation. Unfortunately, the appearance of objects in the real world is usually modeled as a combination of these four mechanisms, which occur simultaneously and are mutually coupled. Thus, physically based models are still used only occasionally, both because of their complexity and that the parameters are not readily available. In addition, since the commercial diffusing components are quite discrepant from the individual supplier, to analyze the complex structures is not necessary.
In this study, we present a methodology, where the measured photometric raw data sets are measured and imported into the diffuse illumination design process. First of all, we will use an experimental way to avoid directly calculating the scattering
characteristics. The measured bidirectional scatter distribution function (BSDF) is applied to characterize the scattering behavior of the diffusing feature. The approach is due to the fact that incident luminance distributions are usually provided numerically by a light-source model or measurement. To construct the outgoing field of the specific diffusing component under a certain illumination by the superposition of these photometry data is the purpose. By means of the proposed process, the required sampling points of an arbitrary diffusing component can be found out. The correlation between the simulated and measured far field radiance distribution would converge to a threshold [14]. In order to confirm the validity of the quantitative algorithm, a commercial 32-inch backlighting source associated with the diffusing sheets for liquid crystal display (LCD) was applied for the demonstration. The conclusions will show that the proposed approach can be used to accurately predict the diffused field of a broad-width source in the laminating systems.
4.2 Energy Integration
In this section, we briefly review the characteristics and terminology of the bidirectional scatter distribution function (BSDF). The associated photometric and geometric quantities in polar coordinates are illustrated in Fig. 4-1, where all the scientific symbols and names through this paper are listed in Table 4-1. In order to simplify the analysis, we restrict the discussion to the transmissive type. Of course, the study can be easily applied to the reflective type without loss of generality. First of all, we recall the bidirectional transmittance distribution function (BTDF) ρ from Chapter 2:
transmitting the specimen. BTDF describes the radiant luminance dLt, which is visible under the angles of observation (θt, φt), induced by the illuminance dEi from a incident-side luminance Li for an incident direction (θi, φi) with a solid angle dωi. Since Li is an available functional description, illuminance E can be decomposed into a linear combination of elementary functions. Equivalently, BTDF can be treated as the two-dimensional impulse response and completely describes the light spreading characteristics of a tested sample. The amount of light transmitted in the outgoing direction can be written as the integral of the BTDF multiplied by the incident flux from each incident direction (θi, φi),
Lt( , )θ φt t ρ θ φ θ φ( , , , ) ( , ) cosi i t t Li θ φi i θ ωid i,
Ω
=
∫
(4.2)where Lt indicates the overall luminance distribution of the transmitted light. Actually, the integration is the aforementioned energy balance equation without the emitted radiance term. Also, the integral can be expressed in a discrete way as following:
(4. the rectangular function rect(ωi.) indicates the solid angle around the specific
i,j i,j
incident angle (θ ,φ ). Based on the linear composition of every j-th components, the hemispherical luminance distributions over the transmission side can be solved accordingly.
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52
Capital letters Greek letters Subscripts
Capital letters Greek letters Subscripts
Fig. 4-1 Photometric and geometric quantities in the polar coordinate
CC correlation coeffi θ, polar c c calculated
W weighting factor τ transmittance e experimenta
ρ bidirectional transm distribution function (sr-1)
Φ light flux j, m count
δ delta functio r referen
.3 Iterative Algorithm
r modeling a diffusing specimen is shown in Fig. 4-4.
The conception is to find the acceptable sampling points of the diffusing component through comparing the outgoing fields by our construction and the reference measurement. The algorithm is started with the aforementioned BTDF measurement.
Based on the measured BTDFs, a rotational superposition was performed to Table 4-1. Nomenclature
4
The proposed procedure fo
implement the rotationally-constructed BTDFs (R-BTDFs). A rotationally symmetric source, that is practically available, illuminated on the sample for the purpose of calculation reference, and the luminance weights at different inclinations of the reference source were introduced into the R-BDTFs to calculate the outgoing field.
After that, we will introduce a merit value (correlation coefficient) to compare the simulated far field luminance with the measured results. The correlation coefficient provides a feedback to correct the sampling point until the merit value converges to an acceptable value. In the following, the sample in Fig 4-3 (b) will be taken for the demonstration of the algorithm.
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4.3.1 Rotational Construction by BTDFs
calculating the transmitted field by con
Fig. 4-2. Modeling procedure for a commercially available diffusing
Because the procedure is based on
structing the BTDFs, the rotational sampling of the BTDF is performed first. A rotationally symmetric field is assumed to illuminate the specimen. Therefore, the dependence along the azimuthal direction can be degenerated by taking convolution
between the measured BTDFs ρj, which is the j-th inclination set as shown in Fig. 4-2, and a comb function along the azimuthal direction:
0
indicate the shifts along φ direction of the j-th inclination. ρjR
is the j-th rotationally-constructed BTDF (R-BTDF). In our case, the width of the horizontal band Δθ is 10 degrees, which is mentioned in the measured results in section 2.2.
Here the sampling number Mj, which means 2π is equally divided into Mj divisions, is increased with the outer band, so the sampling interval Δφj is varied with different j.
Fig. 4-5 shows an numerical calculation of convolution of the 40°-inclination (j=4 the fourth ring) BTDF. Because of the nature of the integrated solid angle, the arrangement of Mj should be directly proportional to the zonal constant [23], which is a convenient factor to calculate the luminous flux emitted into a narrow band and multiplying the summation by a solid angle factor. After a number of straight manipulations, eight donut-like R-BTDF ρjR
can be obtained accordingly.
Fig. 4-5 The convolution of 40°-inclination
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The scattering from a diffusing sheet can be treated as the radiant flux emitted from a secondary flat source. The zonal constant integration technique is a convenient method to numerically integrate the radiation pattern of the light source into a value of the luminous flux. It calculates the luminous flux emitted into a narrow band and multiplying the summation by a solid angle factor, called the zonal constant. We apply this factor into our algorithm, where the sampling point of the angular divisions Mj is proportional to the zonal constant.
The value of zonal constant Cz(θj) can be derived by calculating the surface area for one horizontal band on the surface of the hemisphere and converting the surface area value into the subtended solid angle. Fig. 4-6 (a) schematically shows the surface area of the j-th horizontal band. If the hemisphere is equally divided into J vertical divisions, the polar angular increment Δθ is equal to π/2J. Thus, the zonal constant Cz is equal to the simplified solid angles of the horizontal band:
( ) 2 sin 2 sin .
z j 2 j j
C J
θ = π⋅ π θ = π θΔ θ (5)
However, the continuous solid angle ω of the horizontal band can be derived as
2
Comparing the zonal constant with the exact solid angle of the horizontal segment yields an error as:
2 sin
In our case, the Cz approximation gives an error about 0.25% when the width of the horizontal band is 10 degrees. Fig. 4-6 (b) shows the numerical comparison between the zonal constant and the continuous solid angle. The zonal constants represent the relation of the solid angles on different horizontal band. To make the reconstructed
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arrangement of impulse response in a homogeneous distribution, Mj should be directly proportional to the zonal constant.
(a) (b)
Fig. 4-6 (a) The schematic illustration of the horizontal band. (b) Comparison between exact solid angle and the zonal constant of horizontal band.
4.3.2 Weighting & Superposition of R-BTDFs
A rotationally symmetric reference light source is essential in the calibration of the algorithm. Usually, a Lambertian field is adopted due to its uniform luminance and easily offered by the conventional sources, which include the surfaces of fluorescent lamps or the light diffused by a thick diffusing plate. In our case, the measurement is performed by the setup where the light emitted from a tungsten lamp passes through a diffusion plate and Lambertian emission is measured by conoscopic system. In order to calculate the transmitted field under the reference illumination by BTDFs, each R-BTDF (j=0~7) is weighted by the factor Wj in accordance with the Lambertian reference source subject to the corresponding inclinations. The weights Wj
is the luminance at j-th inclination θj of the reference incident function Li,r over the value in normal direction θ0 under a constant azimuth φ0.
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The scattering light field is determined by the superposition of the eight weighted R-BTDFs. Normalized luminance distribution of the transmitted light can be represented as:
where A is the normalized coefficient, and calculated by
1 , ,
Similar to Eq. (4.3), Eq. (4.6) has an identical form expect the weighting factors Wj are introduced by the reference Lambertian source. Here the cosine term is a tilt factor between the light source and the illuminated plane. Fig. 4-7 exhibits the individual R-BTDFs and their superposition results, which means the constructed outgoing distribution. Although the normalized luminance distribution through the diffusing specimen from a Lambertian source is obtained, however, the accuracy of the calculation highly depends on the number of discrete sampling component.
Consequently, an additional merit index is required to gauge the accuracy of the proposed algorithm. The amount of sampling number is sequentially increased and correlated with its measurement. In the absence of continuity, the algorithm would obtain the discrete points for BTDF superposition and keep the outgoing field in certain accuracy.
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Fig. 4-7 The cross-section of individual 1D-BTDF and summation under Lambertian illumination.
4.3.3 Correlation Coefficient
Correlation coefficient is a merit value to gauge the accuracy of the numerical construction and served as a feedback for correcting the sampling points. Thus, we correlate the calculated field with its measurement, which are both the response of the reference Lambertian source. The correlation coefficient CC is defined as:
where Lc and Le are the normalized luminance distributions of the calculation and its measurement, respectively. Lc and Leare the mean values of both corresponding datasets. If CC is below a threshold value T, it means the calculated field has a certain amount of discrepancy with the real one. Because the deviation is mainly due to the finite sampling points in the superposition, we would increase the sampling number ΣMj by an incremental rate ΣMj×CC-1 until CC is above the desirable threshold value.
Eventually, the amount of the angular sampling ΣMj along the azimuthal direction can
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converge to an acceptable value. In case of the commercially available diffuser in the Fig. 3(b), the correlation coefficient can be achieved upon 98% as the sampling number exceeds 357. Certainly, the sampling number is related to the optical complexity and dependent on the optical features by case. Because the retrieved output field is calculated by the superposition of finite number of BTDFs, the finite sampling points easily cause the discrete calculated fields or incorrect results. The profiles of BTDFs directly affect the required sampling number. The broader BTDF profile requires less sampling points, and the distribution is related to the structure of the diffusing sheets, as we mentioned in section 2.2.
4.4. Evaluation
Aforementioned procedure is employed to determine the sampling grid of the BTDFs for calculating the optical response of the specimen, so we are able to directly apply the results to calculate the transmitted luminance distribution from assigned illuminating sources Li. The weighting function W (j,m) corresponding to the known sampling grid, can be obtained by:
Thus, the transmitted distribution function can be calculated by two-dimensional superposition of the BTDFs multiplied by the corresponding weight function,
( , ) , cos , ( , ).
In addition to the normalized luminance distribution, the absolute value is able to be calibrated by the transmittance τ (θ) as following:
where the Φt and Φi are the overall luminous flux from the incident and transmitted sides of the diffusing sheet. In most case of diffusing sheets, the transmittances τ (θi) are a constant with respect to different incident angles, so the absolute value of the transmittance can be applied on light sources with variant angular distributions. The transmittances can correct the normalized luminance distribution to absolute luminance value. The calculated result for the sample we mentioned is shown in Fig.
4.8(a), where the transmitted luminance distribution was from a commercially available 32”-TV backlighting source. Comparing with the experimental results in Fig 4.8(b), the close agreement with the measurement (CC =98.6%) demonstrates the validity of the proposed training process and corresponding diffusing model. The negligible deviations at large angles were mainly resulted from the measurement errors attributed by the conoscope distortion.
(a) (b)
Fig. 4-8 Angular luminance distribution transmitted through the diffuser from a 32-inch backlighting source by (a) calculation, and (b) comparison of the cross-sections at φ = 0 and 360 degree, where the CC between two curves is 98.6%.
4.5 Summary
A simple and effective algorithm to model the scattering characteristics of the diffuser is proposed and demonstrated. The major advantage of this study lies in that there is
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no need to formulate the complex physical mechanism of the scattering in a microscopic viewpoint. Instead, as long as a backlighting source (Cold Cathode Fluorescent Lamp or Light Emitting Diode) is available, proposed semi-quantitative algorithm can predict the optical radiance and efficiency with high precision without expense of computational time. The proposed algorithm is based on eight measured BSDFs and a reference Lambertian light source. Additional correlation coefficient is employed to evaluate the model accuracy and eventually converge to a stable sampling parameter. Thus, this modeling scheme only needs the BTDFs and the sampling parameter Mj to characterize one diffusing sheet and it is convenient for designers or factories to build a diffuser database. We successfully demonstrate the validity by using a general backlight source, where calculated emergent luminance distribution is 98.6% close to the measurement. In most case, CCcan achieve the value larger than 98%. The algorithm provides a relatively effective way for diffusing simulation, and is useful for the lighting development in display or luminance application.
4.6 References
[1] M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
[2] L. Tsang, J. A. Kong and K. –H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, New York, 2000).
[3] L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topiics s (Wiley, New York, 2001).
[4] A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, 1994).
[5] A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Wave in Random Media 4, 337-367 (1994).
[6] A. Voronovich, Wave Scattering from Rough Surfaces, 2nd Edition
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(Springer-Verlag, Berlin Heidelberg , 1994).
[7] K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surface,” J. Opt. Soc. Am. 57, 1105-1114 (1967).
[8] B. van Ginneken, M. Staveridi and J. J. Koendrik, “Diffuse and specular
[8] B. van Ginneken, M. Staveridi and J. J. Koendrik, “Diffuse and specular