Chapter 4 Calculation Methodology for Energy Balance Equation
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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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 reflectance from rough surface,” Appl. Opt. 37, 130-139 (1998).
[9] K. Tang and R. O. Buckius, “A statistical model of wave scattering from random rough surfaces,” Int. J. Heat Mass Transfer 44, 4095-4073 (2001).
[10] L. Tsang, J. A. Kong, K, -H. Ding and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, New York, 2000).
[11] F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
[12] N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough Surfaces,” Pgys. Rev. Lett. 52, 1798-1801 (1984).
[13] K. Tang, R. Dimenna and R. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surface,” Int. Heat J.
Mass Transfer 40, 49-59 (1997).
[14] E. Kreyszig, Introductory Mathematical Statistics, (Wiley, New York, 1970) [15] M. W. Hodapp, ”Applications for High-Brightness Light-Emitting Diodes” in
Semiconductors and Semimetals Vol. 48, G. B. Stringfellow and M. G. Craford ed., (Academic Press, San Diego, 1997) Semiconductors and Semimetals Vol. 48, Chap. 6, p. 227.
Chapter 5
Application:
Planer Lighting by Remote Phosphor Sheet
5.1 Blue LEDs Array with Remote Phosphor Sheet
In this study, we proposed a blue light excited planar lighting (BLPL) system, which is an alternative LEDs-based direct-emitting planar scheme. Fig. 5-1(b) shows the structure of BLPL, which consists of a blue LEDs array and a YAG-phosphor film.
Here the YAG phosphor is coated on a remote substrate. The YAG-phosphor layer is simultaneously functioned as the diffuser film and wavelength down converter to achieve an ultra-slim LCD backlight application [4].
(a) (b)
Fig. 5-1. Scheme of (a) conventional direct-emitting backlight using white LEDs and (b) BLPL system.
The light emitting mechanism of BLPL system includes wavelength-converting process and scattering by the flat YAG-phosphor particle, as illustrated in Fig. 5-2.
Unlike the conventional phosphor-converted LEDs, the YAG-phosphor in BLPL system is coated on an external substrate by the roll-to-roll coating process [5]. The YAG-phosphor layer here acts as a diffuser film and a wavelength-converter at the same time. As the blue light from blue LED chips irradiate the YAG-phosphor layer,
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portion of the incident blue light will be diffused with high diverging angle, whereas the other part of the incident blue light will be converted to a certain bandwidth, as shown in Fig. 5-3. For the BLPL system, flat phosphor layer redistributes lights and converts the original point light sources to the planar light source. Thus, the proposed optical setup could perform uniform lighting and reduce module thickness of backlight systems.
Because light-emitting mechanism of BLPL involves both spectral and spatial conversion, the traditional ray-tracing computational tools are insufficient to completely treat the underlying physics. Based on the bidirectional photometric measurable data, this paper proposed a methodology to model, analyze and optimize the BLPL system. Finally, a small-size prototype will be demonstrated to validate the model.
Fig. 5-2 The scheme of the BLPL system with light-emitting mechanism.
Fig. 5-3 Spectrum of the incident blue light and the mixed white light.
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5.2 Optical Characteristics of YAG-phosphor
The optical properties of YAG-phosphor layer could be characterized by the measured bidirectional transmittance distribution functions (BTDFs), which is defined as [6-8]: luminance of transmitted light from the sample surface. Here the incident and emit angles are represented by the polar coordinates (θi ,φi) and (θt ,φt), respectively. Since the scattering characteristics of YAG-phosphor layer caused by the randomly distributed phosphor are rotationally symmetric. The measurement and data processing of BTDFs can be simplified by merely considering the variance of polar angle θi with a fixed azimuthal angle φt.
BTDFs represent the angular spread function of the diffusing specimen, which have a variety of optical features due to various manufacturers’ recipes about the refractive index, the size of scattering particles, the density of the phosphor distribution, and so on. 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 LED chip is available, the proposed characterizing method can assist the simulation of the optical radiance and the efficiency.
Differing from the traditional definition of the BTDFs, the BTDFs we defined can represent the scattered light and the emitted light by the spectrally filtered measurements. Owing to the two kinds of optical mechanisms under the blue light illumination, the diffused blue light and the emitted yellow light were measured
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separately. Fig. 5-4 (a) and (b) show the BTDF measured results of the YAG-phosphor.
The angular distribution of the excited yellow light is close to lambertian and has a wider full width at half maximum (FWHM) than the diffused blue light. By these two measured BTDF results, the YAG-phosphor layer could be characterized to develop the BLPL model.
Fig. 5-4 The measured BTDF of (a) the emitted yellow-light radiance and (b) the scattered blue-light radiance
Fig. 5-5 Scheme of theoretical calculation.
5.3 Theoretical Calculation
By using the characterizations of the YAG-phosphor layer, a theoretical model was developed to calculate the transmitted luminance distribution of BLPL system. First of all, the four major parameters of the BLPL configuration should be obtained for the
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theoretical calculation: (1) the BTDF of the YAG-phosphor layer, (2) the intensity distribution Is of the LED chips, (3) the distance h between LED chips and YAG-phosphor layer, and (4) the interval p of the LED arrangement. Through the definition of radiometry, the illuminance E illuminating the YAG phosphor layer at the point (x, y) from the incident direction (θi, φi) of single blue LED can be calculated
Here the geometric relation is schematically shown in Fig. 5-5. Then, the transmitted luminance distribution Lt from the YAG-phosphor layer at point (x, y) can be transferred by the BTDFs
Finally, the total radiating luminance Loutput from the YAG-phosphor layer by LED-array illumination can be calculated by the convolution between the single-LED luminance distribution Lt, (x,y) and a two-dimensional comb function
3 direction, respectively. In this case, the summation and integration were performed by Monte Carlo simulation [9]. In addition to the transmitted luminance distribution, the recycled light which is multi-reflected by the YAG-phosphor film and the bottom reflector can be identically calculated by the bidirectional reflection distribution functions (BRDFs).
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5.4 Simulation
For the LED backlit use, a 5x5 blue LED chips array was placed above a reflector and covered with the YAG-phosphor layer, as shown in Fig. 5-6. We import the measured BSDFs into the commercial software LightTools TM to accomplish the influence of YAG-phosphor on the whole BLPL system. In order to keep the uniformity and luminance as the first merit, the module gap (h) and the interval of blue LED chips (p) were modulated from 4 to 20 mm. Here the luminance uniformity is defined as
,min
which is the luminance ratio of the positions with the minimum luminance and the
which is the luminance ratio of the positions with the minimum luminance and the