Where the light shines: simple DE projectors

[satrya]

Having done a few retrofits and following many discussion threads here, I wanted to better understand how projector optics work. It is worth mentioning some threads that seeded a lot of questions and thoughts for this thread, such as upashi's study of the contribution of different reflector bowl segments to the overall output (link here), and frederik's hypothetical mapping of beam as projected to the wall, to how it lights up the road (link here).

Since I don't work for a company that designs projectors, I'm going to start with the simplest model, a 2-dimensional version of a basic DE (diaxial ellipsoid) projector. I will assume an ideal point source of light, which might be ok for a halogen filament, but not for HID. I realize there are going to be assumptions I make that may be far from the latest state of the art. The derivations are from high school &/ undergraduate level trigonometry and optics, and I will only mention them whenever absolutely necessary. (Hopefully zero times)

This thread will be a work in progress. If you're an industry insider, I'd appreciate some insights.


I will start with the basic shapes.

• To the left, is an ellipse where the reflector bowl will sit.
  • It has two focal points, one marked with a yellow circle, and another marked with a square.
  • A point source of light sits on the yellow circle.
  
  
• To the right, is a circle that represents the shape of the "round" part of the lens.
  • The center of this circle is marked by a "+"
  • The back of the lens is assumed to be completely flat, and the focal point of the lens is marked by an "x", which happens to coincide with the square. The blue arc plots a constant distance from this "x".
  
  

[satrya]

Bowl and lens

Let's add the reflector bowl (red) and the outline of the lens (cyan / light blue). I chose to make the bowl just large enough so that all reflected light will go to the lens. If I had extended the bowl further, some of the reflected light will escape through the gap between the bowl and the lens. Also, I chose to leave a portion on the leftmost area of the bowl open. This will represent the entry point of a bulb. I don't want to cover a side entry bulb at the moment.

I also want to reiterate that I assume a plano-convex lens, meaning one side has a perfectly circular shape, and the other is perfectly flat. Some Hella fog projectors have a flat surface on the inside like this, although none of the low beam / biXenon projector lens I've seen have a flat inner surface.

[satrya]

Let there be light!

On the ellipse's left focal point, assume a point light source generates a uniform light output. This is represented by rays that are spaced equally in terms of their relative angle. In the actual simulation, I will fill the model with more rays (i.e. smaller angle between adjacent rays), but it makes the drawing hard to see.

Some exceptions:

• Light pointing towards the "bulb opening" is assumed to be lost
• Light not touching the reflector bowl will be dealt with later; I imagine the picture is going to get crowded pretty soon, even without these.

It's worth noting that, as seen by the reflector bowl, the spacing of "light" that hits the reflector bowl is not uniform. As you go further from the bulb opening towards the upper and lower edges of the bowl, the spacing between lights get larger and larger. This means that the density of light near the bulb opening is much higher than the density of light near the lip of the bowl (where the mounting flange usually is).

[satrya]

Observations

Given the uneven density of light along the bowl surface, the following is probably true:

• The dimensional tolerance (in terms of maintaining surface that is not wavy) near the bulb opening is more critical than the dimensional tolerance near the lip of the bowl. Any amount of tolerance slop will bounce light farther from its intended course, reducing the ability for the design to focus far. This may impact maximum distance.
• A projector with the same proportions, but with everything made larger, and with the same manufacturing technology (i.e. capital equipment, process control procedures, tech know-how of the staff), is more likely able to achieve the maximum capability of the design. Conversely, making a small projector perform like a bigger projector is more difficult and/or costly.

[satrya]

Reflection of a mirror surface

The rays that reach the bowl will be reflected. Assuming a perfect mirror finish with no scatter or any loss, the reflected ray follows the rule for reflection of rays on a flat surface. (i.e. having equal incident and reflected angles)

Taking the surface equation of the ellipse, the "vertical" direction at any point in the ellipse can be calculated. For those orange rays emanating from the "bulb", the vertical direction of the ellipse surface are drawn as black solid lines. Note how they progressively go further away from the bulb.

[satrya]

Reflected light

The resulting reflected light gathers to the ellipse's other focal point (as marked by the square). Next is what happens to these reflected rays of light as they approach the flat side of the lens. For simplicity, I assume that the lens' focal point (as marked by the "x") coincides with the ellipse's focal point.

[To be continued...]

[satrya]

Continue to the flat side of the lens

Continuing the trace of the rays reflected from the bowl, through the second focal point, and to the flat side of the lens, we get the following distribution:


Note that at the flat side of the lens, the density of light is not even; with the exception of the dead spot at the center due to the lack of reflector bowl (for bulb access), the density (and by extension, the light intensity) decreases as we go further up or down away from the center.

[satrya]

What about the rest of the light?

So far, we've only tracked the rays that are reflected by the bowl. What about the rest? How much of that goes to the lens? In the geometry of this example, adding the rest of the light looks like this.


Note that the rays coming out of the bulb are spaced at equal angles, representing the assumption that the bulb intensity is even across all directions.

In this geometry, about half of the light doesn't get reflected by the bowl. Yet out of that, only a very small fraction, maybe less than 1/4 of the unreflected (i.e. less than 1/8 of the total) reaches the flat side of the lens. The rest (~1/2 - 1/8 = 3/8) goes to the area between the space between the lens and the bowl, which may be covered (like the FX, FX-R, RX330) or partially open like the EvoX-R and Mini D2S 2.0 in the picture below. Of course, the geometry isn't as open as the example above, so the amount of stray light wasted should be somewhat less than 3/8.

[Ilikecars]

Way too smart for me

[satrya]

Way too smart for me

It takes a bit of work, but in the end, the model can give us useful insights into how a projector system works.

[satrya]

Intensity of light at the center of the ellipse

What happens when a vertical screen is placed right at the center of the ellipse (that dashed line right in between the yellow circle and the square)?

Taking more rays (but still following the same rule of each ray spaced at equal angle to each other), we can count the number of rays per unit area, which represents the relative intensity of light. Increasing the rays looks like this:


As we increase the number of rays, we can get a smoother estimate of the relative intensity at the vertical screen placed at the center of the ellipse. As you can expect, the intensity is a combination of reflected light (red lines) and direct light (yellow lines). We can expect that the reflected light will be the dominant source of light, while the direct light will contribute very little, at least close to the center. Here's how it looks:


The direct light (yellow) provides an almost constant intensity along the screen, with the center being the most intense, and gradually going down to zero intensity going all the way up and down. The reflected light (red) on the other hand, has a dead spot in the middle (thanks to the bulb opening), and is most intense right around that opening, and decreases to zero when the surface of the bowl ends.

I suspect a properly designed projector, even when limited to the DE configuration like this, will try to maximize the amount of light that will go through the lens.

[sportyaccordy]

Whoa good work

Couple of questions

Can you see a wall projection of the output of this projector?

Does this program operate in 3D? I.e. could you see the effect of increasing/decreasing the width of the bowl?

Could you see the effect of using a smaller/lower profile light source (i.e. LED chip)?

Good stuff man.

[satrya]

Whoa good work

Thanks

Can you see a wall projection of the output of this projector?

I'm working on it. As you can see, this thread is a work in progress, alongside how far I manage to go through the derivations of the geometry, and how far I manage to go with the code (written in the MATLAB programming software environment). I was debating on whether or not I should post my code and derivation here. I don't mind, except that the thread would be hard to read. I might just post an appendix when everything is done.

Does this program operate in 3D? I.e. could you see the effect of increasing/decreasing the width of the bowl?

The current code is only 2D. I plan to take this as far as I can, and then move on to the 3D. I suspect many things like the effect of putting the shield before, at, or after the second focal point, and how it affects the blue cutoff, can be done using this 2D model. Changing the (2D) aspect ratio of the bowl is also currently possible.

Could you see the effect of using a smaller/lower profile light source (i.e. LED chip)?

That's one of many things I want to "experiment" without having to resort to accumulating parts at home. Right now, the 2D model requires that the source be a point source at the left focal point of the bowl.

As I said, work in progress..

[satrya]

Shape variations

Examples of shape variations, and the fraction of light that goes to the lens (either reflected by the bowl or direct from the bulb). The model parameters are constrained so that the extent of the bowl places all reflected light through the lens. I realize that in an actual low / biXenon / biHalogen projector, this is not necessarily the case. But this provides the minimum bowl and lens material required. Using the edges of the lens is probably not a good thing though.

Standard shape I've been using so far (with more rays illustrated):


With a half sphere lens (like the acrylic ones in the next picture), with the same lens curvature:



With a narrow bowl:


With a thinner lens (and smaller diameter), with the same curvature:

[sportyaccordy]

Thanks man. Sorry to bombard you with questions, I realize these things take time. One other question I have, looking at the diagrams, is what happens to the rays once they pass through the lens? You would think that with a convex lens they would all converge, but I have played with projectors w/o the lens on and something strange happens. W/o the lens (at least on a D2S 2.0 whose bowl is basically the same dimensions in the x and y axis), you basically get a donut with a hole about 15 degrees wide and a donut ring of light from 15 to maybe 40 degrees (from the projector). Obviously when you put the lens on you get a typical projector beam pattern, which means somehow the lens is both converging the beam (to put a hotspot where the hole was) and diverging the beam (to spread the coverage from 40 degrees wide to like 130 degrees wide). Also, somehow it converges the beam vertically and converges the beam horizontally. So in 3D there could be a whole bunch of other variables... i.e. the horizontal and vertical second foci could be in different places; etc etc. I've wanted to do models to try and figure this stuff out but I don't know MATLAB so this is pretty exciting stuff. I wouldn't mind seeing your code... you can probably just post it as a .txt attachment with comments to explain what does what.