Modeling of lamps through a diffuser with 2D and 3D picket-fence backlight models

Transcription

1 Modeling of lamps through a diffuser with D and 3D picket-fence backlight models Richard J. Belshaw*, Roger Wilmott, John T. Thomas** General Dynamics C4 Systems, Ottawa, Ontario, Canada ABSTRACT Laboratory photometric measurements are taken of a display backlight one metre away from the emission surface (diffuser) with a whole acceptance angle on the photometer of about 0.15 degrees (.18mm spot size at emission surface). A simulation method was sought to be able to obtain the brightness uniformity (luminance peak to trough ratio from above one lamp to the null between lamps in a picket-fence backlight). A 3D raytrace BackLight model in TracePro[1] and a D Mathematical model in Matlab[] have been developed. With a specimen backlight in the laboratory, a smooth luminance profile was measured by the photometer on the diffuser surface. Ray Trace models in both 3D and D take too long to produce smooth continuous filled distributions. The Mathematical D approach, although with limitations, yielded smooth solutions in a very reasonable time frame. 1 OBJECTIVE Initially the goal was to be able to measure brightness uniformity (i.e. peak to trough illumination ratios) of different geometrical placements of fluorescent lamps with respect to each other, the back reflector and the diffusive emission surface. The focus of this paper is an examination of modeling techniques applied to picket-fence fluorescent lamp backlights. Light piped backlight modeling will also be addressed briefly. RAY-TRACING.1 Background Ray-tracing is an industry accepted standard for simulating and evaluating the optical design and performance of a given opto-mechanical model (which might involve lenses, diffusers, films, and light sources). A sufficient number rays to effectively sample the distributions and produce a smooth result are required. A sub-technique called importance sampling reduces the necessary number of rays by introducing probability characteristics in the model. If a ray would normally be chosen at a random angle of emission, a separate child ray is emitted towards an importance target. Such a ray would carry a flux proportional to the probability that such a path would be taken, compared to all other possible paths that could be taken from that emission. TracePro, developed for NASA by Lambda Research Corporation, is one of several software packages that provide 3D opto-mechanical modeling; TracePro is actively supported and new and improved releases are issued periodically, often with improvements suggested by the users of the package. TracePro was selected as the backlight modeling tool because it was reputed to be a proven system for backlight modeling.. TracePro model TracePro allows the user to build a solid 3D model, with light sources of specified luminance, and then traces a number of rays from the light sources (number specified by user) which all carry a given amount of flux. Upon a ray striking a surface, a percentage is absorbed, a percentage is diffuse reflected, a percentage specular reflected, a percentage diffuse transmitted (equals zero for pure reflective surfaces), a percentage specular transmitted (and of course for conservation of energy, the total is equal to the amount incident). Diffuse reflected and transmitted rays spawn child rays of appropriate flux. If any of these child rays is lower than a user selected flux threshold they are not re-propagated as active rays. Cockpit Displays IX: Displays for Defense Applications, Darrel G. Hopper, Editor, Proceedings of SPIE Vol. 471 (00) 00 SPIE X/0/$

2 Some solid materials can be selected from an optical library, such as acrylic plastic for example. Surface properties can be applied to any given surface in the model such as reflective white paint, which would have no transmission components at all, and would be mostly diffuse reflective. Refraction of rays into and out of differing transmissive media is carried out as a matter of course. Our intent was to model a photometric scenario, which would mimic the laboratory situation where the photometer was to be placed 1 metre away from the emission surface (diffuser) of the backlight. The photometer would have a mm spot size with respect to the emission surface. Another scenario was also envisioned, in order to minimise the computational load (the close scenario). This scenario constructed an analogue to the original photometer scenario that would place the observation screen 1mm away from the emission surface, but would still have a light admission cone with a mm spot size on the emission surface..3 Ray-tracing problems We observed limitations and experienced difficulty in all attempts to model the picket fence style of backlight. The main obstructions in our attempts to use TracePro were as follows: 1) Photometer modeling implies a very small measurement area places 1 metre away from the light source. As a result, the number of rays incident on this surface is a small proportion of the total modeled for the backlight. ) Ray-tracing with a small number of rays produces grainy/non-smooth results. Increasing the number of rays to produce smooth or acceptable results resulted in long computational runs. Raytrace times for an acceptable smoothness were on the order of 10 days (using a powerful GHz workstation); too long to get the bugs out of the model or to model the impact of design changes (i.e. diffuser modeling, dimensional and other surface parameters). 3) With the close scenario, placing the observation screen 1mm away, smooth, useable results were obtained from about 50,000 rays per lamp (setting the lower flux threshold to or 0.3%), by integrating the resultant map along the length of the lamps, and avoiding integration of edge effects. However, these results can not be said to model the laboratory photometer measurement, but may provide a useful analogue. 4) Diffuser modeling is critical to the overall system model. Characteristic data obtained by measurements 3 of 85% Clarex[3] diffusive films by Astra-Products fit to β (1 + aθ θ + bθ θ + cθ ) quite α 0 0 θ 0 well, and even the simplified cos ( θ θ ) provided an adequate fit to our data. However, the ABg 0 model used by TracePro[1] proved to be extremely difficult to fit to data. If you already know what B and g values you want, it will pick an appropriate numerically integrated value of A. However discovering the value of B and g for the fit is non-trivial. Figure 1 provides a graphical illustration of the problem in curve fitting. The * values represent experimental data and the other curves represent the polynomial (solid), cosine (dash) and ABg (dash-dot) models. Data was obtained by varying the incident angle of a laser light source and measuring the transmitted portion through diffuser material normal to the surface. Note that in Figure 1, the polynomial model with β=1, a = ,b = , and c = , providing an area under the curve of Area =.78. With β = 0.81/0.78, Area =.81; For the Cosine model with α = In the ABg model, the best curve fit was A = 0.1, B = 0.1, and g = The area under the curve represents the amount of percentage of flux that is passed by the model. For example if 1 Lumen is incident on such a surface with Area under the curve=0.81, then 81% of this flux would be passed by the surface on the first incidence of the light. 190 Proc. SPIE Vol. 471

3 These * points indicate measured data Polynomial ABg model Figure 1: Luminance Data vs. Polynomial model (Area=.78)vs. Cosine model vs. ABg model (85% Clarex) We believe that there is a functional shape for diffusers different from ABg or the polynomial model that would be a better characteristic fit to the diffuser data, but these results are adequate for our present modeling purposes. Note that the area under polynomial is also equal to that of the area under the cosine and ABg models when scaled by the same factor β. The polynomial model is a very good fit to this data (but care should be taken in applying any polynomial model due to inherent interpolation difficulties with such a model). The ABg is not far behind, though it is difficult to pick out or solve for the three parameters that will give a good fit. Picking B=.01 for instance yields far less useful results. While the ABg model may have experimental data to back its use for reflective surfaces, transmissive data appears to have a little more than a subtly different functional shape, though not far off for some carefully chosen parameter values. For transmission through a diffuser, one is forced (in TracePro) to use the ABg[1] model for computing Total Scatter (TS) (= = 85% transmission for Clarex[3] diffuser, 4% specular)..4 Diffuser Models The critical scatter property of diffusive surfaces is modeled for 3D and D cases by (see equations 1a) and 1b respectively[1]: π π A TS( 3D) = cos( θ )sin( θ ) dθ dφ H H g φ = 0θ = 0 B + β β 0 π = ' A TS(D) cos( θ ) dθ θ = 0 B + sin( θ ) g Equation (1a) is the 3D version and (1b) represents the D version for normally incident light, and: (1a) (1b) Proc. SPIE Vol

4 A or A is a parameter that scales the overall integral to obtain the correct amount of total scatter. B serves two functions: a) It prevents the denominator from going to zero. b) It, in conjunction with A, uniquely determines the peak value of TS for any beta difference, and the shape of the curve near zero degrees. c) It determines the steepness also, by determining how far out from zero degrees the value remains near peak value (backwards s shape). The final factor, g determines the steepness of the curve (although the overall ABg model has a characteristic peak shape, roll-off and attenuation characteristic that is fundamentally a different shape from the cosine or polynomial functions mentioned above). Some D-curve fitting to the data was done to establish g for both the 75% and the 85% Clarex[3] diffuser types..5 Ray-tracing results Once characteristic values had been established and entered into TracePro, simulations were carried out and the peak to trough (p/t) ratios were compared for various different angle filters and diffuser types. Angle filters are mathematical creations that pass light energy that is within some cone angle of incidence, and filters out all other light energy whose angle of incidence is greater than the nominal incidence angle value. Angle filters are useful in mimicking a photometer measurement over an entire surface, in that a photometer measurement has a specific spot size (and hence allowable angle of incidence) for any specific measurement. For an angle filter of 3.08 degrees and no diffuser (100% transmission), the peak to trough ratio predicted by TracePro was approximately 1, which matched the p/t ratio from our laboratory measurements (made with mm diameter measurement spot size). The p/t prediction, however, for 85% Clarex[3] was between 1.39 and 1.51, very different from our laboratory measurements of and respectively, and not a reliable value. The difference between a p/t ratio of near one, and that near 1.5 is huge. It indicates a systemic modeling problem. The situation was that either a different angle for the angle filter was needed (which would throw out the 100% transmission test case; p/t=1), or a change in the g values (steepness) for the model was needed (which had already been fit to experiment). Neither prospect made sense. Clearly the model was not true to experiment, indicating that a new method was required. Preferably one that took less than 3 hours per run, and that was truer to the photometer scenario (i.e. observation screen located 1 metre away from emission surface)..6 Interim conclusion After much effort, we concluded that it was not possible to obtain a derivable 3D model which would produce experimentally verifiable results either in a reasonable period of time or not. We think this is primarily due to the underlying ray-tracing technique this particular backlight problem is just VERY hard for a ray tracing technique to solve). A new method was searched for that would allow us to reach our stated objectives. 19 Proc. SPIE Vol. 471

5 3 D MATHEMATICAL APPROACH: EFFECTIVELY IMPLEMENTING AN INFINITE NUMBER OF RAYS 3.1 Background In view of the problems associated with use of TracePro in a 3D backlight modeling scenario, we returned to fundamental questions. Instead of ray-tracing, might there not be another method that would allow us to approximate the peak/trough results? 3. Development of a D ray-tracing model In an effort to produce smooth data results within reduced computation time, we simplified the model to a D model (a planar slice through the 3D model, lengthwise). This model contained: One back-reflector (L=0mm long, 94% diffuse reflective, 5% specular, 1% absorptive); Two circular lamps a distance yc from the back reflector (cross sectional diameter 4.1mm, L=0mm between centres, 1 Lumen each); One diffuser a distance D from the back reflector (either 75% Clarex or 85% Clarex using the ); and experimentally fitted polynomial function: poly ( ) θ θ 0 An observation plane (line). The observation line was placed a distance K=1000mm (1 metre) away from the emission surface (line) (behind an angle filter line, that mimics a mm diameter spot size at the diffuser). See Figure and Figure 3. Back reflector Diffuse Light Lamp Direct Light x Diffuser 1metre separation Angle Filter x3 Observation Screen Figure : Components of Model. L yc D x K x3 Figure 3: Dimensions of Model Proc. SPIE Vol

6 3.3 D ray-tracing results Despite this simplification, the D ray-tracing model in Matlab[] still took 4 hours computation time (compared to 10 days for the comparable photometric 3D model in TracePro), and still did not produce very smooth results. This lead to the question: how might the luminance profile at the diffuser be calculated without ray-tracing? 4 DEVELOPING A MATHEMATICAL MODEL 4.1 Background to mathematical model In view of the difficulties experienced with TracePro and the limited success with a D ray-tracing model. we decided to attempt a purely mathematical solution in D, effectively delivering an infinite number of light rays from the lamps. The fundamental requirements from this model remain a need for accuracy and acceptable computational speed. An important part of this model is to mimic the photometer measurement aspect. 4. Implementation Referring to Figure 3, let (0<x<L) be some point on the diffuser, and let (0<x3<L) be some other independent point on the observation screen. What light might be incident upon point x (i.e. direct and diffuse) from all possible sources inside the backlight? How might the diffuse light from the diffuser be incident upon a point x3 at the observation screen (taking the angle filter into account)? It was decided to make a series of approximations by using 1 st order effects only. Picking a point x on the diffuser, consider the light sources incident on that point. There would be direct light from both lamps (at their respective incidence angles; these angles are important for calculating diffusion effects with the cosine or polynomial functions identified above). There would be a small amount of specular reflected light from the rear reflector, and there would be diffuse light from the entire back reflector, as well as diffuse light reflected from the diffuser First order approximations The following approximations were made: Approximation 1: Ignore the sides of the box for now, that is a nd order effect. Approximation : Ignore the reflected light off the lamps from the back reflector, which is a nd order effect. I.e. assume that the lamp sources are infinitesimal point sources. Approximation 3: Assume diffuse light from the back reflector has a lambertian characteristic when emitted from the diffuser. Approximation 4: Assume that the 15% of the light that is reflected from the diffuser (for an 85% transmissive Clarex[3]), is recirculated and re-emitted as diffuse lambertian light from the diffuser. 4.. Calculating the amount of light at a point x on the diffuser: In the following analysis (please refer to Figure 3): G The Back reflector is at zero height. G Lamp centres are located at yc above the back reflector. G The diffuser located at a distance D above back reflector. G Lum is the flux emitted by each lamp. G L= 0mm is equal to distance of lamp centre from lamp1 centre. The signal path direct light on the diffuser at point x (0<x<L) is: 194 Proc. SPIE Vol. 471

7 From lamp1: Lum fa( x) = ; θa = atn( D yc, x) π ( x) + ( D yc) And from lamp: Lum fb( x) = ; θb = atn( D yc, x L) π ( L x) + ( D yc) () (3) The specular reflected light from the back reflector (only 5% specular, representing the 0.05 factor in equations) is: From Lamp1: fc( x) = π And from Lamp: fd( x) = π Lum*0.05 yc* x D + yc + ( yc) yc* x + D* L L + D + yc Lum*0.05 x* yc ; θc = atn D, x D + yc ( yc) x* yc + D * L ; θd = atn D, x D + yc (4) (5) The diffuse component from Lamp1 via a 94% diffusive back reflector is: fe( x) = L x 0.94* Lum + yc π D + ( x x= 0 (π ) x) dx (6) x being some point (0<x<L) along the back reflector. And the diffuse component from Lamp is: ff( x) = L ( L x) 0.94* Lum + yc D + ( x x= 0 (π ) π x) dx (7) Now we have the inputs to the diffuser. Note that all are functions of x. Imagine we now have a point x3 on the observation screen (1 metre away from the diffuser = K = 1000mm (all distance units are in mm i.e. millimetres)). And let π θ = 0 β poly ( θ ) dθ = Area (8) Proc. SPIE Vol

8 For 85% Clarex, For 75% Clarex, Area =.7 Solve for β..81 β =,Area= In the following expression for L(x3), let: θ = atn( K, x3 x) (9) Let x3 indicate the location of a point on the observation screen such that 0<x3<L. Then, the Light incident at x3 is given by (85%: spec =.04; 75%: spec =.05): L L( x3) = x= 0 where : poly( x) = β fa( x)* C * poly( θ θa) fb( x)* C * poly( θ θb) + + π K + ( x3 x) π K + ( x3 x) fc( x)* C * poly( θ θc) fd( x)* C * poly( θ θd) + + dx π K + ( x3 x) π K + ( x3 x) ( fe( x) + ff( x)) (1 Area spec)** Lum + K ( x3 x) L K ( x3 x) π + π + 3 ( 1+ a x + b x + c x ) piecewise continuous. This integral is basically a series of path integrals (along x) producing a result along x3. (10) 4..3 Solution Since the photometer has a spot size of mm at 1000mm away,θ has a window of *atan(1mm/1000mm) = degrees total aperture. In actual fact the whole angle for the photometer is 0.15 degrees. The last term in the L(x3) expression is a DC offset term for the diffuse light that will be re-emitted from the diffuser in a lambertian distribution (equal probability of emission at every angle from the diffuser surface). We are assuming here for example, that if the Clarex film is 85% transmissive, then only 15% of the initial light will be re-emitted by the diffuser as diffuse light on a second or later pass (this is a macroscopic approximation). We are also assuming that the diffuse light from the rear reflector, is emitted from the diffuser as diffuse light. All the other terms are moderated by the poly() distribution which smoothes the specular nature of those rays, but gives greater precedence in its distribution to flux that is oriented in the same direction as θ. NB: Angular differences greater than cos( π π π θ θ i > )=cos( )=0). Since normally θ is a very small, and the i difference should always be π or less. are treated as equal to a difference of θ are between 0 and π (i.e. π,their 196 Proc. SPIE Vol. 471

9 4.3 Limitations of the mathematical model The goal was to obtain a 1 st order approximation of backlight luminance uniformity. The approximations that were made (in order of importance) are: 1) No sides were included in the model, though they can be at little extra computational cost. It would actually be advantageous to have two models, one for the central brightness uniformity simulation (as we have carried out here), and a separate lamp model with sides, to evaluate the edge effect separately. ) The lamps were assumed to be infinitesimal point sources. The major ramification of this approximation is that there would be no reflections off the lamp surfaces from the sides or the back reflector prior to incidence on the diffuser. A separate surface calculation might be done to determine the brightness of the sides and back reflector (assuming continuous illumination) and to measure the amount of specular and diffuse illumination of the lamp surfaces. A further computation, assuming that the fluorescent surface of the lamp would emit as a lambertian reflector, could be carried out to calculate the subsequent amount of light that would be incident on the diffuser from these reflections. This would probably be small, on the order of 5%, and is considered a second order effect. 3) For the total incident amount of light, say for example with the 85% diffuser, a full 15% would be reflected back into the backlight. One assumption here is that somehow this 15% will make its way back again to the diffuser, over and over again, until it is spent. The second assumption here is that this 15% makes its way back to the diffuser as diffuse light (i.e. from all and any angle in a lambertian manner), and is reemitted ultimately as diffuse light (again lambertian) to the outside world from the diffuser. This acts as a Direct Current or DC offset in determining the illuminance of the observation screen. This is a sort of macroscopic modeling and makes sense, but has not been verified experimentally (though the overall peak to trough result has been verified). 4) Assume that diffuse light emitted from the back reflector, or from any sides, has a lambertian characteristic. We can change the model from lambertian to ABg or some other model, but a white painted surface is likely to be quite close to lambertian in the 1 st order. 5) It is realized that there would be some reflections that would continue for some time between the sides, back reflector and the lamp surfaces before ever encountering the diffuser. This effect has been assumed to be negligible compared to the major effects (see 3 above for the treatment of this fact macroscopically). 5 RESULTS Table 1 presents the backlight luminance uniformity for both measured and predicted data. Table 1: Results from modeling Peak/Trough data summary Diffuser Type Measured.18mm spot size, Whole angle: 0.15 TracePro ( close - 1mm) 3.08 filter (whole angle). Mathematical ( close 1mm).4 filter (whole angle) Mathematical (1 metre) mm spot size. Whole Angle: (poly model) 1.1(poly model) 75% Clarex N/A (poly model) 85% Clarex to (poly (g=1.9356) model) 100% (clear) (no g term) NA 0.7(α=5000, cosine model) Effectively no diffuser present. Proc. SPIE Vol

10 Note that due to time-limited termination of the computation run, data is not presented for the 3D photometric TracePro option. Please see Figure 4 and Figure 5 below, which present the experimental data: Luminance Plot of CTD Backlight with 75% Clarex Diffuser (measured) 650 Luminance (fl) Distance (cm) Figure 4: Photometric measurements (1 metre) - 75% Clarex Diffuser, p/t= Luminance Plot of CTD Backlight plus 85% Clarex Diffuser (m easured) Luminance (fl) Distance (cm) Figure 5: Photometric measurements (1 metre) - 85% Clarex Diffuser, p/t=1.138 Clarex 85%-Centre two of eight lamps Rays; ABg model (g=1.9356) Figure 6: TracePro ( close - 1mm) - 85% diffuser (g=1.9356): p/t=between 1.39 to Proc. SPIE Vol. 471

11 In Figure 6, we show data obtained from TracePro[1] for the close-in observation pane. The observation screen is located just 1mm away from the 85% Clarex[3] diffuser. The peak/trough ratio = between 1.39 to 1.51 (there is ambiguity due to the coarseness of the ray tracing technique); originally there were 8 lamps included in this computation, but Figure 6 was taken from the middle of these results to minimise the side or edge effects. The 8 lamp result was obtained in 3. hours with TracePro[1] (18 vertical sample points per 160 mm, integrated over 100 horizontal points = 1,800 sample points; plotted data is from 1600 points). Figure 7: D Mathematical Model; 85% Clarex; close - 1mm away. Half angle = 1. ; Peak/trough =1.131 Figure 7 presents results from the D Mathematical model (for just two lamps). The observation screen is located 1mm away from diffuser surface, and the filter acceptance whole Angle =.4 degrees (half angle = 1. degrees): The peak/trough result = (polynomial model). Note the fine-grained smoothness of this result (081 points per 0 mm). This result was obtained in about 9 minutes on a GHz workstation with interpreted Matlab[], compared with 3. hours for a comparable plot from ray-tracing in TracePro[1] (16 points per 0mm, integrated 100 times; 1600 points total). Figure 8 presents results from the D Mathematical model (two lamps). The observation screen is one metre away from diffuser surface, and the filter acceptance half angle = degrees. The peak/trough result = 1.1 (polynomial model). Notice the similarity in functional form between Figure 7 and Figure 8. This lends some credence to the hypothesis that the close scenario should indeed be a useful analogue to that of the photometer scenario for peak/trough estimates. 6 DISCUSSION 6.1 General We have characterised a sample picket fence backlight by measurement and compared these results to theoretical results in order to validate different model solutions. The modeling was implemented in three different ways because the initial ray-tracing method implemented in TracePro, was found to be unrealistically computationally intensive. It was also intended to accommodate different types of backlight incorporating light-pipes. This part of the project is presently incomplete. Proc. SPIE Vol

12 Figure 8: D Mathematical Model; 85% Clarex; 1 metre separation. Half angle = ; Peak/trough=1.1; mm spot size 6. Modeling picket-fence backlights Our intention was to use an off the shelf ray-tracing package to obtain a general-purpose indicator of backlight uniformity. This was found to be impossible with a ray tracing technique such as that fundamentally employed by TracePro. Simulation was initiated using a model that mimicked a backlight with a photometer placed 1 metre in front of it. The model effectively moved the photometer in both x and y-axes to build up a full picture of the luminance map. Clearly, rays that are reflected or transmitted with diffuse characteristics spawn additional rays with lesser intensity but the number of such rays increases rapidly as the reflections and transmission process is implemented. However, the number of rays impinging on the photometer is small, since the acceptance surface is miniscule at a distance of 1 metre. Ray tracing is still a preferred technique for many optical problems, but this particular problem is not one of those. To reduce the computational effort, rays, which were not considered important by virtue of their directional properties, and rays which fell below a significant energy threshold were culled. Still, the number of rays required to build a useful picture of the backlight uniformity in dimensions involved unrealistically long runs, even with a high-end fast PC. To reduce the computation time, it was decided to adapt the 3D ray-tracing model to measure the total flux impinging on a small area of the backlight surface. In this way, we hoped to gather most of the rays resulting from the calculations describing the internal components of the backlight. Our reasoning for this approach was that the total flux energy exiting the diffuser was of interest to us, since all of it would be utilised in the display. We found that only by restricting the acceptance angle of the viewing surface, could we obtain a luminance profile that matched experimental data. We could not find a filter that produced a luminance profile matching all three reference measurement cases. Neither could we justify changing the acceptance angle of the filter to suit the different scenarios. A detailed examination of the model used within TracePro to simulate diffusing surfaces leads us to believe that the ABg model used here does not exactly match our observations of typical diffuser characteristics, although it may be adequate. It was also impossible to modify the basic characteristics of the model, despite the fact that a relatively simple polynomial model appeared to more accurately simulate our diffusers. This conclusion was reached after implementing curve-fitting solutions to data gathered from two different diffusers using a laser to provide a collimated light source and measuring the transmitted and reflected patterns over angle. 00 Proc. SPIE Vol. 471

13 Since the ray-tracing model did not provide the required uniformity data, we turned to a mathematical model. This requires that the basic layout assumptions be simplified. However, we have found that by simulating a photometer layout, it is possible to obtain an accurate match to the reference data for all three cases used in this project. The resulting computational time is relatively very small compared to ray-tracing, but the drawback lies in giving up the flexibility of the ray-tracing approach. Clearly, the mathematical approach is less of a generalised model than a particularized case model. It appears possible however, that this may be improved. In any case, the mathematical model does provide essential data required to optimise a backlight design. 6.3 Light-pipe based backlights The ray-tracing approach appears fully applicable to modeling light-piped backlights. However, we anticipate much the same type of problem from the modeling tool, especially in the area of diffuser characteristics. Regarding the applicability of mathematical techniques to a light-piped backlight, a -D approach may be sufficient. Such a model would separate the problem into two components. One, running along the line of the light source in a horizontal plane feeding the light-pipe would accommodate edge effects and the possibility of multiple individual sources such as LED s. The second component would model the luminance of the exit surface at right angles to the light source (i.e. vertical plane). A true 3-D approach may also be possible but is clearly a complex implementation task. 7 CONCLUSIONS The mathematical model provides very quick answers (9 minutes compared to from 3. hours to 10 days). The results are finer grained (higher quality) and essentially give a result as would be obtained by an infinite number of rays in ray-tracing (the distributions are amply filled as opposed to being sparse ). The mathematical model uses numerical integration and does not simulate some second order effects. Ray-tracing is not a realistic option for brightness uniformity simulation of picked-fence style backlights; other techniques, such as the D Mathematical approach, provide better, more timely answers. Ray-tracing appears, however, to be the logical choice for modeling 3D light pipes, though the computational time may be very long. The mathematical approach may also be applied to light-piped backlights, but will require additional development. ACKNOWLEDGEMENTS General Dynamics Canada wishes to acknowledge the support of the Defence Industrial Research Programme, and Mr. Douglas Laurie-Lean, who administers this programme, without whose assistance, this work would not be possible. REFERENCES [1] TracePro is a trademark product name of Lambda Research Corporation. Importance Sampling is a technique that TracePro uses. [] Matlab is a trademark product name of the MathWorks Inc.. [3] Clarex DR III C 75% transmissive and 85% transmissive white diffusive filters by Astra-Products. [4] Optical Scattering: Measurement and Analysis. John Stover, S P I E-International Society for Optical Engineering (December 1995). ISBN: *richard.belshaw@gdcanada.com; phone ;fax ;http// General Dynamics Canada, 3785 Richmond Rd, Nepean, Ont., Canada, KH 5B7. **john.thomas@gdcanada.com; phone ; (see * above for fax, website and address detail). Proc. SPIE Vol