Measuring and Modelling BSDF of LEDs and Diffusers

Before we continue our journey on polarization, we need to build ourselves a homogeneous, polarized, light screen also known in optics as a backlight. Because at The Pulsar we like well-operated engineering, we will take the occasion to optimize our backlight through a simulation model :)

Our backlight will be composed of an array of LEDs with a diffuser plate as shown in Figure 1. Other backlight arrangements exist but the one of Figure 1 is fairly typical and is considered as a state-of-the-art starting point when you need relatively powerful illuminations.

Our backlight is therefore composed of two main elements: (a) LEDs which act as light sources, and, (b) a diffuser plate which acts as a homogenizer element. To correctly engineer our backlight, we must therefore understand how we can model each of them to correctly account their effects in the simulation. This is precisely what this post is about.

If you look at Figure 1, you will see that each LED emits light rays in a range of directions, with different intensities, and that each of these rays then emits secondary rays at the diffuser plate, with varying intensities as well. We therefore need to understand what is the angular distribution of intensities emitted by the LED and what is the angular distribution intensities scattered by the diffuser plate for a given input angle.

To measure the angular distribution of ray intensities emitted by the LED, I made the setup of Figure 2 which is available [∞] here for reproduction for all our [∞] Premium members. The system is composed of a rotation stage, a collimated lens, a pinhole and a photodiode.

We saw in one of our [»] previous posts that placing an aperture at the focal point of a lens will convert ray slopes at the lens into vertical deviations at the aperture. The setup of Figure 2 is a direct application of this principle because the rays emitted by the LED can hit the lens at any height, but they will reach the detector only if they pass through the aperture. Since the pinhole is very tight, only rays parallel to the optical axis will pass through. And since the LED is mounted on a rotation stage, we can record the amount of light that reaches the photodetector for varying angle positions and map the LED angular distribution (also called angular spectrum). The setup Figure 2 requires a high-quality photodiode amplifier circuit, such as the one we made [»] here.

As an illustration, I mapped the angular spectrum of Thorlabs M625L4 LED in Figure 3 and compared it to a cosⁿ model.

The cosⁿ model is extremely useful when modeling LEDs as it uses only information that can be found in LED datasheet while offering satisfactory performances for most common LEDs, including both broad and narrow angular spectrum ones.

As its name suggests, the cosⁿ model has the form of a cosine raised to some power n

where n can be fitted from experimental measurements or be computed from the angle where the intensity drops by half

This angle is usually provided in the datasheet as twice θ_h under the name “viewing angle” or any equivalent naming convention. Note that the model of Figure 3 was drawn using the value of 80° provided in Thorlabs datasheet (θ_h=40°) and provide an error as low as 3% rms with the experimental values. We can therefore either measure a LED, or assume a given value of n for our simulations based on the LED datasheet.

Modeling our diffuser isn’t very different from modeling our LED although we need a different formula than the cosⁿ one because, this time, the behavior is affected by the direction of the incoming rays. There exist many different models, all belonging to the generic naming of bidirectional scattering distribution function (BSDF).

A popular empirical BSDF among optical softwares is the ABg model:

where A, B and g are model parameters of the diffuser plate and β, β₀ are the components of the unit vectors that lies in the diffuser plane for the scattered direction and the incoming ray direction respectively.

When both rays lie in the same plane, the model reduces to

with θ₀ the angle of the incoming ray relative to the diffuser plane normal.

If the incoming ray is perpendicular to the diffuser (θ₀=0), the model further simplifies to

One of the interesting features of ABg models, not demonstrated here, is that it is possible to scale the A and B parameters when changing incoming monochromatic wavelengths:

where the subscripts 2 and 1 corresponds to the initial model parameters and to the adapted model parameters respectively.

Another, more historically interesting, feature is the ability to fit the ABg model directly from a logarithmic plot. Indeed,

and when θ is large enough such that B becomes negligible,

Here, however, I’m using Matlab to fit ABg experimental data using a [»] least-square approach.

The modified setup to measure diffusers is given in Figure 4. A collimated laser is used as a source in place of the LED of Figure 2 but the measurement principle stays the same. As for the LED setup in Figure 2, it is important that the emission point (the diffuser plate) is coincident with the pivot axis of the rotation stage. All CAD files are also included in the package provided [∞] here.

An experimental measurement of the ABg model of Thorlabs ground glass diffuser DG10-220 is given in Figure 5 along with a typical measurement provided by Thorlabs on a different sample. There is good agreement between the acquired data and both the data provided by Thorlabs as well as the ABg fit with an rms error of 4% between the model and the experimental data. Most of the error is due to what seems to be happening on-axis with the diffuser (θ≈0°).

The ground glass diffuser of Figure 5 is however not desirable as a backlight because the diffusion cone is too small (the intensity drops too quickly). Here, I opted for a Clarex DR-III 85C acrylic diffuser, for which the experimental data as well as the fit is provided in Figure 6. You can see that the diffusion cone is much larger this time. Note as well that the data is noisier because the light from the laser is scattered much more than with the narrower ground glass diffuser, leading to overall lower light intensities on the photodiode. Nonetheless, the fit is extremely satisfactory on a 120° diffusion angle although the extreme tail doesn’t seem to be modeled correctly by the ABg model.

In terms of methodology, it is interesting to point out that it was beneficial to start measuring the Thorlabs diffuser for which data was available from Thorlabs. Only with the confidence that our setup was measuring properly that we could attempt at measuring a real, unknown, sample.

Also, due to the price of the Clarex diffuser, I initially tried to go for cheaper Chinese diffusers bought on Amazon but the results where not satisfactory at all. There was a lot of grain on the cheap diffuser, it melted during machining, and it blocked a lot of the incoming light. You can check a picture of the two diffusers side-by-side in Figure 7.

This closes our first part of the backlight article series. We now have all the tools required to accurately simulate an existing LED and diffuser element, such that we can simulate and optimize our backlight in the next post!

Do not hesitate to share your thoughts on the [∞] community board to let me know if you enjoyed this post!

I would like to give a big thanks to Sebastian, Alex, Stephen, Lilith, James, Jon, Jesse, Karel, Zach, Kausban, Michael, Sivaraman, Samy, Shaun, Onur, Sunanda, Benjamin, Themulticaster, Tayyab, Marcel, Dennis, M, Natan and RottenSpinach who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!