Calculating model of light transmission efficiency of diffusers attached to a lighting cavity

doi:10.1364/OE.18.006137

Calculating model of light transmission efficiency of diffusers attached to a lighting cavity

Ching-Cherng Sun, Wei-Ting Chien, Ivan Moreno, Chih-To Hsieh, Mo-Cha Lin, Shu-Li Hsiao, and Xuan-Hao Lee »View Author Affiliations

Optics Express, Vol. 18, Issue 6, pp. 6137-6148 (2010)
http://dx.doi.org/10.1364/OE.18.006137

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Abstract

A lighting cavity is a reflecting box with light sources inside. Its exit side is covered with a diffuser plate to mix and distribute light, which addresses a key issue of luminaires, display backlights, and other illumination systems. We derive a simple but precise formula for the optical efficiency of diffuser plates attached to a light cavity. We overcome the complexity of the scattering theory and the difficulty of the multiple calculations involved, by carrying out the calculation with a single ray of light that statistically represents all the scattered rays. We constructed and tested several optical cavities using light-emitting diodes, bulk-scattering diffusers, white scatter sheets, and silver coatings. All measurements are in good agreement with predictions from our optical model.

1. Introduction

Lighting and display are one of the most important branches of technology in the beginning of the XXI century. In lighting, the impact is from the growth of solid-state lighting device such as light emitting diodes (LEDs), which enable more color saturation, life time, design freedom and environmental benefit. However, owing to the point-source nature and high luminance of the LED, much glare occurs when the optical design does not address eye care [1

J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86(6), 363–370 (2003). [CrossRef] [PubMed]

T. Kasahara, D. Aizawa, T. Irikura, T. Moriyama, M. Toda, and M. Iwamoto, “Discomfort glare caused by white LED light sources,” J. Light Vis. Env. 30(2), 95–103 (2006). [CrossRef]

]. This is usually solved by enlarging the effective area of the light source. There are many ways to increase the emitting area [3

J. A. Wheatley, G. J. Benoit, J. E. Anderson, R. W. Biernath, D. G. Freier, T. R. Hoffend, C. D. Hoyle, T. T. Liu, J. D. Lu, M. A. Meis, V. V. Savvateev, C. R. Schardt, M. E. Sousa, M. F. Weber, and T. J. Nevitt, “Efficient LED light distribution cavities using low loss, angle-selective interference transflectors,” Opt. Express 17(13), 10612–10622 (2009). [CrossRef] [PubMed]

A. Travis, T. Large, N. Emerton, and S. Bathiche, “Collimated light from a waveguide for a display backlight,” Opt. Express 17(22), 19714–19719 (2009). [CrossRef] [PubMed]

]. A simple, low-cost, and widely used method is to place the light sources into a cavity covered with a diffuse translucent sheet. The diffuser scatters the transmitted light, and reflects a significant fraction of the incident light back into the cavity, eventually homogenizing the spatial light distribution. Figure 1 shows some examples of lighting cavities assembled with LEDs behind a diffuser plate. The diffuser spreads the optical flux across a larger area so that the LEDs cannot be seen by an observer and the glare effect is reduced. Figure 1(b) shows an example where one diffuser is applied to an LED luminaire. A large cavity with an LED array behind the diffuser also allows light painting of ceilings [5

I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008). [CrossRef]

Fig. 1 (a) A simple lighting cavity, with and without diffuser. (b) An example of LED luminaire with and without a covering diffuser sheet. (c) A direct LED backlight (of a television display) without diffuser.

In addition to lighting, the light source enlargement also is employed in liquid crystal display (e.g. television, laptop, and monitor), where the backlight component transforms a set of line or point light sources into a plane light source as large as the screen size. In backlight technology, a low cost approach that allows high-dynamic range is called direct-view backlighting [5

I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008). [CrossRef]

–8

H. F. Chen, T. H. Ha, J. H. Sung, H. R. Kim, and B. H. Han, “Evaluation of LCD local-dimming-backlight system,” J. Soc. Inf. Disp. 18(1), 57 (2010). [CrossRef]

]. In such a case, a diffuser instead of a light guide plate is the key component. In a direct backlight, a diffuser covers the chamber that contains the light sources, e.g. an LED array. An open chamber of a direct LED backlight is shown in Fig. 1(c), which in operation is covered with a diffuser.

Although lighting and display are different topics, both have a common demand, to keep the optical efficiency as high as possible. The general way to manage the optical power of a lighting cavity covered with diffusers (LCCD) is to make a simulation with ray tracing program using a very large amount of rays [9

M. Gebauer, P. Benoit, P. Knoll, and M. Neiger, “P-9: Ray Tracing Tool for Developing LCD-Backlights” SID Symposium Digest of Technical Papers–May 2000–Volume 31, Issue 1, pp. 558-561

]. However, the scattering model of diffusers is complex [10

C. H. Tien and C. H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17(2), 723–732 (2009). [CrossRef] [PubMed]

], the diffuser properties may vary from one to another manufacturer, and many optical parameters of the diffuser and the optical cavity should be known so that the cavity simulation becomes very difficult and time consuming. Then the usual way to get the optical efficiency is the experimental measurement [6

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. 16(4), 519 (2008). [CrossRef]

]. This is why a practical method to calculate the optical efficiency is demanded.

The balance between light extraction efficiency and illumination uniformity or glare comfort of the LCCD relies heavily on the overall light transmission of the diffuser. In other words, the diffuser attached to a lighting cavity (DALC) is the dominant factor of the LCCD optical efficiency. In this paper, we present a simplified optical model to calculate the transmission efficiency of a DALC. Section 3 presents the equations to compute the overall transmission efficiency. In Section 4 the model is demonstrated by several experimental measurements by using bulk-scattering diffusers. Section 5 shows how the cavity walls and source placement influence the light extraction efficiency. Before explaining the model, we would like to describe the optical cavity structure in the next section.

2. Optical cavity with diffusers

There may be a wide variety of cavity shapes, but the squared chamber is the most popular [3

A. Travis, T. Large, N. Emerton, and S. Bathiche, “Collimated light from a waveguide for a display backlight,” Opt. Express 17(22), 19714–19719 (2009). [CrossRef] [PubMed]

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. 16(4), 519 (2008). [CrossRef]

–11

D. Voigt, I. A. Hagendoorn, and E. W. M. van der Ham, “Compact large-area uniform colour-selectable calibration light source,” Metrologia 46(4), S243–S247 (2009). [CrossRef]

]. Therefore, we consider the basic LCCD to be a box coated with reflecting films [see Fig. 2(a) ]. Typically, an optical cavity is covered with one or two diffusers, and the light sources are located on the bottom plane. The cavity has four reflective sidewalls, i.e. except the light sources and the diffuser all the other surfaces are coated (or covered) with reflective film. This enables the light reflected back to be incident on the diffuser again through multiple reflections and then the overall transmission efficiency of the DALC increases.

Fig. 2 (a) Optical cavity with 1 and 2 diffusers. (b) Diffuser plate. R₀ and T₀ are the single-shot power reflection and transmission efficiency at normal incidence, respectively. Here Φ_in is the input light flux at normal incidence, Φ_T is the total transmitted light flux to the right of diffuser, Φ_R is the total reflected light flux to the left of diffuser. ϕ_n and ϕ_m are the light fluxes associated to each ray of light reflected and transmitted, respectively.

We consider that the diffuser is a non-structured scattering plate, i.e. its optical properties randomly scatter the incident light rays [12

M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, and J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15(22), 1261–1263 (1990). [CrossRef] [PubMed]

]. In the practice, the transmission and reflection properties of randomly scattering diffusers are not ideal [12

–15

B. Chevalier, M. G. Hutchins, A. Maccari, F. Olive, H. Oversloot, W. Platzer, P. Polato, A. Roos, J. L. J. Rosenfeld, T. Squire, and K. Yoshimura, “Solar energy transmittance of translucent samples: A comparison between large and small integrating sphere measurements,” Sol. Energy Mater. Sol. Cells 54(1-4), 197–202 (1998). [CrossRef]

]. For example, the transmitted light through a diffusing plate is a mixture of two angular radiation patterns (a direct and a diffuse component), and the direct-diffuse ratio increases as a function of wavelength [13

P. Manninen, P. Kärhä, and E. Ikonen, “Determining the irradiance signal from an asymmetric source with directional detectors: application to calibrations of radiometers with diffusers,” Appl. Opt. 47(26), 4714–4722 (2008). [CrossRef] [PubMed]

–15

]. This effect is large at near-infrared wavelengths, but low at the visible range [13

,14

P. Manninen, “Characterization of diffusers and light-emitting diodes using radiometric measurements and mathematical modeling,” Doctoral Dissertation Thesis, Helsinki University of Technology, Finland (2008).

]. Another non-ideality of random diffusers is that the center of the angular distribution of the transmitted light depends on the angle of incidence of light (see Appendix and references [10

C. H. Tien and C. H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17(2), 723–732 (2009). [CrossRef] [PubMed]

,12

]). Because we are considering the total extracted flux (integration of the angular radiation pattern) in the visible range, these non-idealities have little effect in the total efficiency. This is why we use the single-shot transmission and reflection efficiency in our analysis [Fig. 2(b)]. The term “single-shot” refers to the behavior of one beam of light that interacts only one time with an optical surface.

The light sources can be arranged in a variety of configurations to achieve spatially uniform emission of light from a backlight or luminaire. The placement of sources inside the LCCD may strongly influence the illumination uniformity, but slightly influences the overall light extraction. If LEDs are used as the light sources, the divergence angle of the LED will decide the thickness of the cavity for the uniformity issue [6

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. 16(4), 519 (2008). [CrossRef]

,16

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]

]. In general, a thick LCCD is needed for narrow beam LEDs, and a thin cavity is associated with wide beam LEDs. The enlargement of LED divergent angle through first-level (package level) optical design usually causes the degradation of luminance (lm/m²sr) from the cavity. Therefore, in many cases when considering the effect of thickness, energy efficiency, uniformity, optical design and assembling way, it makes sense to use two diffusers in a cavity. Generally, more scatterings of light cause more uniformity and smaller thickness of the cavity, but also cause lower luminance. Thus, a heavy-doped diffuser or two light-doped diffusers is/are used in a thin cavity to achieve high uniformity [16

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]

Once we have described the LCCD structure, we proceed to estimate the flux transmission efficiency of the DALC in the following section.

3. Light transmission efficiency

The optical transmission efficiency of the DALC is the ratio of the output luminous flux using diffuser to the output luminous flux without diffuser [Fig. 3(a) ]. The complexity of the scattering theory and the difficulty of the multiple calculations involved, make intractable the exact computation of the optical efficiency of a DALC. We overcame these problems by carrying out the calculation with a single light ray that is representative of all the scattered rays. Then we obtain a simple approximation but very close solution rather than the exact but very complex answer. A similar approach is widely used in the theory of integrating spheres, where the radiation exchange within a spherical enclosure of diffuse surfaces simplifies to a single ray of light [17

Labsphere, Inc., A Guide to Integrating Sphere Theory and Applications, at http://www.labsphere.com/

,18

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).

]. The theory analyses the multiple reflections of a single ray inside the integrating sphere. This ray is representative of all the scattered rays because the fraction of light flux that it transports from one point to another is independent of the incidence angle.

Fig. 3 (a) Defining the optical efficiency of the diffuser incorporated into the cavity. (b) Multiple reflections of the equivalent ray of light inside the chamber incorporated with one single diffuser. Here the T is the one-shot transmission efficiency of the diffuser plate; the R is the one-shot reflection efficiency of the diffuser; and R_b is the one-shot reflection efficiency of the inner surfaces.

Here, the key idea is to consider only one ray of light instead all the scattered rays. Due to the statistical nature of the scattering process in the diffuser and internal walls, the equivalent ray must be representative of the average. Therefore, in order to deduce the efficiency equation we use a single ray that is incident at an equivalent angle of incidence. The calculation of the effective angle is described in the Appendix. For example, if the scattering power of the inner walls is low (for example the silver coatings used in Sections 4 and 5), and if the LEDs used have a Lambertian radiation pattern (typical of high power LEDs), the analysis shows that the effective angle is ~45º. But if the internal walls show strong scattering (for example white scatter sheets), the effective angle of incidence reduces to ~30º due to the multiplication of scattering events.

Taking into account this simplification we calculate the optical efficiency for a single equivalent ray of light. The multiple reflections involved, make the computation to be a sum. As shown in Fig. 3(b), the optical efficiency of the DALC is

η = T + T R_{b} R + T R_{b}^{2} R^{2} + \cdot \cdot \cdot = \frac{T}{1 - R_{b} R},

(1)

where T and R are the one-shot transmission and reflection efficiency of the diffuser, respectively. And R_b is the reflection efficiency of the other surfaces in the cavity. Note that T, R, and R_b must be measured at the equivalent angle of incidence. Also note that absorption is implicitly included in this calculation, and then not only T but also R must be experimentally measured. For example, the one-shot absorption of the diffusers used in our measurements can be deduced from the sum of T and R measurements shown in Fig. 11 in the Appendix.

Fig. 11 Single-shot transmittance (left) and reflectance (right) of diffusers D55, D60 and D70 in function of the angle of incidence.

In the case of two diffusers, first we have to consider the reflected lights between the two diffuser plates [Fig. 4(a) ]. As shown in Fig. 4(b), the transmission (T₁₂) and reflection efficiency (R₁₂) of the two-diffuser system are

T_{12} = T_{1} T_{2} (1 + R_{1} R_{2} + R_{1}^{2} R_{2}^{2} + \cdot \cdot \cdot) = \frac{T_{1} T_{2}}{1 - R_{1} R_{2}},

(2)

R_{12} = R_{2} + T_{2}^{2} R_{1} (1 + R_{1} R_{2} + R_{1}^{2} R_{2}^{2} + \cdot \cdot \cdot) = R_{2} + \frac{T_{2}^{2} R_{1}}{1 - R_{1} R_{2}},

(3)

where T₁ (T₂) is the one-shot transmission efficiency of the first (second) diffuser, and R₁ (R₂) is the one-shot reflection efficiency of the first (second) diffuser. The optical efficiency of the two diffusers attached to the lighting cavity can be expressed as

Fig. 4 Multiple reflections in a cavity with two diffusers. (a) Multiple reflections of the equivalent ray of light between the two diffusers. (b) Multiple reflections of the equivalent ray of light inside the chamber incorporated with two diffusers. T₁₂ is the one-shot overall transmission of the 2 diffusers, i.e. the summation of transmissions shown in (a).

η = \frac{T_{12}}{1 - R_{b} R_{12}} .

(4)

Again we note that T₁, T₂, R₁, R₂, and R_b must be measured at the effective angle of incidence.

It is simple and easy to use Eqs. (1) and (4) to calculate the overall transmission efficiency of a DALC. We illustrate their simplicity, and experimentally validate their applicability in the following section.

4. Experimental comparison

For the purpose of demonstration, we assembled and tested a wide variety of lighting cavities with LEDs inside. We used two kinds of reflective sheets for the sidewalls: silver scatter sheet and white scatter sheet (see Fig. 5 ). These sheets are usually employed in both lighting and display backlighting. The cavity size was 9×9×4 cm³, and contained a square array of 2×2 white LEDs. When using two diffusers, one diffuser was located at half of cavity, and the other at the top. We used a small cavity because of two reasons: to show the edge effects (reflections at side walls), and to facilitate the introduction of the cavity inside the integrating sphere for testing.

Fig. 5 Cross section of inner walls and diffuser for experimental measurements. (a) Shows the cross-section of the silver scatter sheet. (b) Shows the cross-section of the white scatter sheet. (c) Bulk-scattering diffuser plate.

In our experiments, the diffuser plate is a bulk-scattering diffuser (BSD). In such diffusers many optical particles are randomly suspended throughout the plastic plate to scatter the incident light rays, see Fig. 5(c). To cover the cavity we used three types of BSDs, which are numbered as D55, D60 and D70. The manufacturer states that the corresponding single-shot power transmission efficiencies T₀ (at normal incidence) are 55%, 60% and 70%, respectively.

In order to evaluate Eq. (1) and Eq. (4), we need the effective one-shot transmission and reflection efficiency (measured at oblique incidence) of BSDs, i.e. T and R. Figures 6(a) and 6(b) show the experiment setup we used to measure these effective efficiencies. The BSD sample was 2×2 cm², and it was attached with black paper to block unwanted light contributions. Although the scattering profile of a diffuser changes in function of the wavelength over the visible range [13

,14

], the variation of single-shot transmittance and reflectance is small. For example, the change over the visible spectrum of the T with respect to T(λ=532nm) is 2.6%, 2.4%, and 3.8% for D55, D60 and D70, respectively. The sensitivity of the human eye has its peak in the green color, and then for measurements we used a green laser as a representative wavelength of the visible spectrum. We used a large integrating sphere (SphereOptics 40-inch diameter integrating sphere photometer).

Fig. 6 Experiment setup with an integrating sphere for measuring the optical efficiencies of diffuser, side walls, and LCCD. (a) Shows the set up for measuring the effective one-shot transmission coefficient T. (b) Shows the set up for the effective one-shot reflection coefficient R. The angle of incidence of all measurements is at 45 degrees when using silver coatings, and it is 30 degrees when using white scatter sheets. (c) Experiment setup for measuring DALC efficiency, η.

When using silver coatings for the inner walls, the measurement was performed at a 45º angle of incidence. The measurement angle was 30º when the LCCD was assembled with white scatter sheets. The effective one-shot reflection efficiency of inner walls, R_b, was measured at oblique incidence in the same way as shown in Fig. 6(b). Note that although the measurement set up of Fig. 6(b) cannot avoid some multiple reflections between the sample and the sphere, this problem is minimized by using a small sample and a large integrating sphere.

The measurement setup we used to measure the transmission efficiency of DALC is shown in Fig. 6(c). The comparison between the theory and experimental measurements for LCCDs assembled with one and two BSDs is shown in Fig. 7 . Despite the differences between assembled cavities and the ideal one, calculations and experiments are in quite good agreement for the twelve LCCDs that we tested. The deviation between the calculation and experimental results is within 4.96% for LCCDs that use white scatter sheets, and it is within 4.7% for LCCDs with silver coating sheets.

Fig. 7 Comparison between theory and experiment. In graphs “Cal” is the value given by Eqs. (1) and (4), and “Exp” indicates the experimental measurement. The graphs show the efficiency η of bulk scattering diffusers attached to a lighting cavity. Some cavities are assembled with one diffuser (1D) and others with two diffusers (2D). The inner walls of cavities in graph (a) are white scatter sheets, and the inner walls of cavities in plot (b) are silver scatter sheets.

Let us illustrate one efficiency prediction by using Eqs. (1) and (4). We can note of these equations that the reflectivity of the inner surface, R_b, is quite important to the cavity efficiency. Figure 8 shows η vs. R_b for a cavity with one diffuser. This plot suggests that the use of reflective coatings having an effective reflectance exceeding 96% could give a light transmission efficiency as high as 92%.

Fig. 8 Efficiency of DALC with one diffuser in function of the effective reflectivity of inner walls.

5. Effects of cavity height and LED pitch

The diffuser not only works in combination with back reflectors, but also with the lateral reflecting walls of the optical cavity. Light that is reflected back into the cavity is recycled by all the reflective walls of the cavity. Therefore, the pitch between LEDs and the height of the chamber influence the overall optical efficiency of LCCD. Although Eqs. (1) and (4) do not take into account the sidewall interaction, its effect is in general small for the optical efficiency of DALC. Figure 9 shows a comparison between the calculated efficiency η and the measured values for several LCCD configurations. Fig. 9(b) shows that the largest deviation is 7.3 (~10% difference). Despite the physical differences between an ideal and an assembled cavity, the largest deviation is low because we are comparing the theoretical calculation with the experimental measurement of 24 different cavities. In addition, considering that the cavity is relatively small, the deviation between the calculation and measurements is low. The difference is mainly due to side wall effects, then this deviation should become lower as the side walls become shorter and the bottom wall becomes larger.

Fig. 9 Effect of LED pitch P, and height of cavity walls H. This figure shows the calculated values (●) by using Eqs. (1) and (4), and experimentally measured values (▲,■). These graphs are for LCCDs assembled with silver scatter sheets.

6 . Summary

There are several approaches to convert a set of bright point-like light sources to a larger extended light source. But optical diffusers, used in conjunction with an optical cavity, are the most popular solution in many lighting and display applications. In lighting, the glare of bright point-like sources is reduced by transforming them to a much larger glowing lamp with less glare and softer brightness. In displays, the spatial uniformity of the screen brightness is increased by transforming the point-like sources to a larger extended emitting source.

Both display backlighting and general lighting have a common demand, to keep the optical efficiency as large as possible. However, it is impractical to analyze the light extraction efficiency of a lighting cavity covered with diffusers (LCCD) because of the complexity of the optical process. Therefore, the usual method to determine the optical efficiency is the experimental measurement. Considering that the efficiency of the diffuser attached to a lighting cavity (DALC) is the dominant factor of the overall efficiency, we developed a simplified optical model to calculate the light transmission efficiency of a DALC. We overcame the complexity of the scattering theory and the difficulty of the multiple calculations involved, by carrying out the calculation with a single light ray that is statistically representative of all the scattered rays. The optical model was demonstrated by several experimental measurements. We constructed and tested several LCCDs by LED arrays, bulk-scattering diffusers, white scatter sheets, and silver coatings. Despite the differences between assembled cavities and the theoretical LCCD, theory and experiment were in good agreement. The deviation between the calculation and experimental results was within 4.96% for LCCDs assembled with white scatter sheets, and within 4.7% for LCCDs with silver coating sheets.

Appendix: Effective angle

In principle, one can compute an approximate solution of the light transmission efficiency via only one ray of light instead all scattered rays. It is based on the assumption that an effective angle of incidence can be deduced. We outline the development of such an equivalent angle approach in this appendix. We derive an equation to calculate the effective angle in function of the type of LEDs and diffusers that assemble the lighting cavity.

An important simplification is to consider only one ray of light instead all the scattered rays inside the cavity. We use an equivalent ray that effectively represents the average of the multiple scatterings between the diffuser and the internal walls. For this issue we evaluate a weighted mean. In this type of mean each of the values does not contribute equally to the total average, i.e. some values weight more than others [19

D. Terr, “Weighted Mean” From MathWorld-A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/WeightedMean.html

]. We use the weighted arithmetic mean of angles θ_i to get the effective angle θ_eff

θ_{e f f} = \frac{\sum_{i} w_{i} θ_{i}}{\sum_{i} w_{i}},

(5)

or for continuous values

θ_{e f f} = \frac{\int w {(θ)}_{} θ_{} d θ}{\int w {(θ)}_{} d θ},

(6)

where w_i (or w(θ)) are the weighting factors. Angles with a high weight contribute more to θ_eff than do angles with a low weight. Therefore, the key issue is to find the correct weighting factor. We use the transmitted (or reflected) flux of light as weighting factor with which each angle of incidence may be weighted in proportion to its degree of effect on the light flux transmitted (or reflected). This makes physical sense because the effective angle is used to calculate the light transmission efficiency of the DALC.

Because the light extraction efficiency of the lighting cavity relies heavily on the diffuser, we only consider the light transmitted (and reflected) through the diffuser plate. Fig. 10 shows a cone of light emitted by one LED source. This narrow beam is incident on the diffuser at an angle θ. For practical purposes an LED can be approximated as a directional point source [20

C. C. Sun, W. T. Chien, I. Moreno, C. C. Hsieh, and Y. C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009). [CrossRef] [PubMed]

], then a differential of transmitted light flux through the diffuser is approximately

d Φ_{T} = I (θ, ϕ) T (θ) d Ω = I (θ, ϕ) T (θ) \sin θ d θ d ϕ,

(7)

and a differential of reflected flux is

Fig. 10 Cone of light flux, emitted by an LED, transmitted through a diffuser plate.

d Φ_{R} = I (θ, ϕ) R (θ) d Ω = I (θ, ϕ) R (θ) \sin θ d θ d ϕ .

(8)

Here I(θ,φ) is the radiant (or luminous) intensity (W/sr or lm/sr) of the LED [21

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]

], where θ is the polar angle, and φ is the azimuthal angle. T(θ) and R(θ) are the single-shot transmittance and reflectance at an angle of incidence θ on the diffuser plate.

By choosing w(θ)=dΦ/d θ as the weighting factor, the effective angle (in transmission) is

θ_{e f f - T} = \frac{\int_{0}^{2 π} \int_{0}^{\frac{π}{2}} I (θ, ϕ) T (θ) θ \sin θ d θ d ϕ}{\int_{0}^{2 π} \int_{0}^{\frac{π}{2}} I (θ, ϕ) T (θ) \sin θ d θ d ϕ} .

(9)

And the effective angle for the reflected light is

θ_{e f f - R} = \frac{\int_{0}^{2 π} \int_{0}^{\frac{π}{2}} I (θ, ϕ) R (θ) θ \sin θ d θ d ϕ}{\int_{0}^{2 π} \int_{0}^{\frac{π}{2}} I (θ, ϕ) R (θ) \sin θ d θ d ϕ} .

(10)

For practical purposes we chose to use only one effective angle for all measurements. The effective angle we use is then

θ_{e f f} = \frac{θ_{e f f - T} + θ_{e f f - R}}{2} .

(11)

The radiation pattern of most high power LEDs is rotationally symmetric about the optical axis [21

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]

], i.e. I(θ,φ)= I(θ). Then, using the discrete notation, the effective angle we used in Setions 4 and 5 is

θ_{e f f} = \frac{1}{2} [\frac{\sum_{n} I (θ_{n}) T (θ_{n}) θ_{n} \sin θ_{n}}{\sum_{n} I (θ_{n}) T (θ_{n}) \sin θ_{n}} + \frac{\sum_{n} I (θ_{n}) R (θ_{n}) θ_{n} \sin θ_{n}}{\sum_{n} I (θ_{n}) R (θ_{n}) \sin θ_{n}}] .

(12)

In Section 4 we present several experimental measurements by using bulk-scattering diffusers (BSDs). We used three types of BSDs, which are numbered as D55, D60 and D70. Their single-shot transmittance and reflectance in function of the angle of incidence is shown in Fig. 11. We used the measuring set up shown in Figs. 6(a) and 6(b). Using these functions, T(θ) and R(θ), and using I(θ)= I₀ cosθ as the intensity function of LED, the effective angle θ_eff is 45.28º, 44.98º, and 45.5º for the BSDs D55, D60 and D70, respectively. Therefore we chose θ_eff=45º for experiments. This is particularly valid when the scattering power of the inner walls is low (for example silver coatings, Fig. 12(a) shows its angular distribution in transmission). If the inner walls have a strong scattering power, the light trapped inside the cavity can find an easier way to escape from the cavity [22

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174 (1993). [CrossRef]

]. The multiplication of scattering events results in a reduced effective angle of incidence. For example, if the internal walls are white scatter sheets [Fig. 12(b)] shows its radiation pattern), the effective angle of incidence reduces to ~30º.

Fig. 12 Angular radiation pattern of the inner walls for different angles of incidence. (a) Shows the silver coating, and (b) white scatter plate.

Acknowledgement

This study was sponsored by the National Science Council with the contract no. 97-2221-E-008-025-MY3.

References and links

1.	J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86(6), 363–370 (2003). [CrossRef] [PubMed]
2.	T. Kasahara, D. Aizawa, T. Irikura, T. Moriyama, M. Toda, and M. Iwamoto, “Discomfort glare caused by white LED light sources,” J. Light Vis. Env. 30(2), 95–103 (2006). [CrossRef]
3.	J. A. Wheatley, G. J. Benoit, J. E. Anderson, R. W. Biernath, D. G. Freier, T. R. Hoffend, C. D. Hoyle, T. T. Liu, J. D. Lu, M. A. Meis, V. V. Savvateev, C. R. Schardt, M. E. Sousa, M. F. Weber, and T. J. Nevitt, “Efficient LED light distribution cavities using low loss, angle-selective interference transflectors,” Opt. Express 17(13), 10612–10622 (2009). [CrossRef] [PubMed]
4.	A. Travis, T. Large, N. Emerton, and S. Bathiche, “Collimated light from a waveguide for a display backlight,” Opt. Express 17(22), 19714–19719 (2009). [CrossRef] [PubMed]
5.	I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008). [CrossRef]
6.	C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. 16(4), 519 (2008). [CrossRef]
7.	M. Trentacoste, W. Heidrich, L. Whitehead, H. Seetzen, and G. Ward, “Photometric image processing for high dynamic range displays,” J. Vis. Commun. Image Represent. 18(5), 439–451 (2007). [CrossRef]
8.	H. F. Chen, T. H. Ha, J. H. Sung, H. R. Kim, and B. H. Han, “Evaluation of LCD local-dimming-backlight system,” J. Soc. Inf. Disp. 18(1), 57 (2010). [CrossRef]
9.	M. Gebauer, P. Benoit, P. Knoll, and M. Neiger, “P-9: Ray Tracing Tool for Developing LCD-Backlights” SID Symposium Digest of Technical Papers–May 2000–Volume 31, Issue 1, pp. 558-561
10.	C. H. Tien and C. H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17(2), 723–732 (2009). [CrossRef] [PubMed]
11.	D. Voigt, I. A. Hagendoorn, and E. W. M. van der Ham, “Compact large-area uniform colour-selectable calibration light source,” Metrologia 46(4), S243–S247 (2009). [CrossRef]
12.	M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, and J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15(22), 1261–1263 (1990). [CrossRef] [PubMed]
13.	P. Manninen, P. Kärhä, and E. Ikonen, “Determining the irradiance signal from an asymmetric source with directional detectors: application to calibrations of radiometers with diffusers,” Appl. Opt. 47(26), 4714–4722 (2008). [CrossRef] [PubMed]
14.	P. Manninen, “Characterization of diffusers and light-emitting diodes using radiometric measurements and mathematical modeling,” Doctoral Dissertation Thesis, Helsinki University of Technology, Finland (2008).
15.	B. Chevalier, M. G. Hutchins, A. Maccari, F. Olive, H. Oversloot, W. Platzer, P. Polato, A. Roos, J. L. J. Rosenfeld, T. Squire, and K. Yoshimura, “Solar energy transmittance of translucent samples: A comparison between large and small integrating sphere measurements,” Sol. Energy Mater. Sol. Cells 54(1-4), 197–202 (1998). [CrossRef]
16.	I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]
17.	Labsphere, Inc., A Guide to Integrating Sphere Theory and Applications, at http://www.labsphere.com/
18.	R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).
19.	D. Terr, “Weighted Mean” From MathWorld-A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/WeightedMean.html
20.	C. C. Sun, W. T. Chien, I. Moreno, C. C. Hsieh, and Y. C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009). [CrossRef] [PubMed]
21.	I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]
22.	I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174 (1993). [CrossRef]

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(230.3670) Optical devices : Light-emitting diodes
(080.4295) Geometric optics : Nonimaging optical systems
(080.4298) Geometric optics : Nonimaging optics

ToC Category:
Geometric Optics

History
Original Manuscript: December 24, 2009
Revised Manuscript: February 13, 2010
Manuscript Accepted: March 5, 2010
Published: March 11, 2010

Virtual Issues
Focus Issue: Solar Concentrators (2010) Optics Express

Citation
Ching-Cherng Sun, Wei-Ting Chien, Ivan Moreno, Chih-To Hsieh, Mo-Cha Lin, Shu-Li Hsiao, and Xuan-Hao Lee, "Calculating model of light transmission efficiency of diffusers attached to a lighting cavity," Opt. Express 18, 6137-6148 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6137

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References

J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86(6), 363–370 (2003). [CrossRef] [PubMed]
T. Kasahara, D. Aizawa, T. Irikura, T. Moriyama, M. Toda, and M. Iwamoto, “Discomfort glare caused by white LED light sources,” J. Light Vis. Env. 30(2), 95–103 (2006). [CrossRef]
J. A. Wheatley, G. J. Benoit, J. E. Anderson, R. W. Biernath, D. G. Freier, T. R. Hoffend, C. D. Hoyle, T. T. Liu, J. D. Lu, M. A. Meis, V. V. Savvateev, C. R. Schardt, M. E. Sousa, M. F. Weber, and T. J. Nevitt, “Efficient LED light distribution cavities using low loss, angle-selective interference transflectors,” Opt. Express 17(13), 10612–10622 (2009). [CrossRef] [PubMed]
A. Travis, T. Large, N. Emerton, and S. Bathiche, “Collimated light from a waveguide for a display backlight,” Opt. Express 17(22), 19714–19719 (2009). [CrossRef] [PubMed]
I. Moreno, “Creating a desired lighting pattern with an LED array,” Proc. SPIE 7058, 705811 (2008). [CrossRef]
C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, “Brightness management in a direct LED backlight for LCD TVs,” J. Soc. Inf. Disp. 16(4), 519 (2008). [CrossRef]
M. Trentacoste, W. Heidrich, L. Whitehead, H. Seetzen, and G. Ward, “Photometric image processing for high dynamic range displays,” J. Vis. Commun. Image Represent. 18(5), 439–451 (2007). [CrossRef]
H. F. Chen, T. H. Ha, J. H. Sung, H. R. Kim, and B. H. Han, “Evaluation of LCD local-dimming-backlight system,” J. Soc. Inf. Disp. 18(1), 57 (2010). [CrossRef]
M. Gebauer, P. Benoit, P. Knoll, and M. Neiger, “P-9: Ray Tracing Tool for Developing LCD-Backlights” SID Symposium Digest of Technical Papers–May 2000–Volume 31, Issue 1, pp. 558-561
C. H. Tien and C. H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17(2), 723–732 (2009). [CrossRef] [PubMed]
D. Voigt, I. A. Hagendoorn, and E. W. M. van der Ham, “Compact large-area uniform colour-selectable calibration light source,” Metrologia 46(4), S243–S247 (2009). [CrossRef]
M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, and J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15(22), 1261–1263 (1990). [CrossRef] [PubMed]
P. Manninen, P. Kärhä, and E. Ikonen, “Determining the irradiance signal from an asymmetric source with directional detectors: application to calibrations of radiometers with diffusers,” Appl. Opt. 47(26), 4714–4722 (2008). [CrossRef] [PubMed]
P. Manninen, “Characterization of diffusers and light-emitting diodes using radiometric measurements and mathematical modeling,” Doctoral Dissertation Thesis, Helsinki University of Technology, Finland (2008).
B. Chevalier, M. G. Hutchins, A. Maccari, F. Olive, H. Oversloot, W. Platzer, P. Polato, A. Roos, J. L. J. Rosenfeld, T. Squire, and K. Yoshimura, “Solar energy transmittance of translucent samples: A comparison between large and small integrating sphere measurements,” Sol. Energy Mater. Sol. Cells 54(1-4), 197–202 (1998). [CrossRef]
I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006). [CrossRef] [PubMed]
Labsphere, Inc., A Guide to Integrating Sphere Theory and Applications, at http://www.labsphere.com/
R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).
D. Terr, “Weighted Mean” From MathWorld-A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/WeightedMean.html
C. C. Sun, W. T. Chien, I. Moreno, C. C. Hsieh, and Y. C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009). [CrossRef] [PubMed]
I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008). [CrossRef] [PubMed]
I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174 (1993). [CrossRef]

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