Angles Of Reflection

If you have been around projected displays for very long, you have probably at least heard of the possibility that one area of a screen might be visibly brighter than the rest of it.  That brighter area is a “hot spot” and while it is usually fairly easy to detect, it is not always so easy to predict.  There are some very useful guidelines that will help in assessing the risk of a hot spot appearing but only if they are properly understood and applied.  What does it take to determine if the audience will see a:

Hot Spot or Not?

As you may suspect, the answer to this question is going to involve looking at a gain chart and measuring some angles.  We have talked about half gain angles on several occasions before and how important they are for brightness perception so it stands to reason that they will be involved as well.  In addition to these points, I would like to focus on two important components of uniformity issues that have received a little less exposure: bend angles and reflectivity.

Bend angles, in this context, are simply the angles between a single ray of light and a corresponding line between the viewer’s eye and the origin of that ray.  The rays we are interested in do not originate at the projector, as might be expected, but rather at each point on the screen where a projected ray is reflected. 

Figure 1’s top down views illustrate these elements, here selecting two points on the edges of the screens visible to the viewer where two rays are reflected and thus showing two bend angles for each.  The dashed lines represent the path from those points to the viewer’s eyes and the dotted lines are the rays of light reflected by the screen at those points.  Please note that, in the interest of keeping the illustrations free from extraneous information, the lines showing the projected rays are not included.

Of course, a real display will have nearly infinite angles and the lines will be drawn through space in three dimensions instead of only two.  The corner opposite the viewer will usually represent the widest angle in reality but for the purpose of demonstration, we can assume that the edges at the horizontal level of the viewer’s eyes will be the maximum.

Now that we have this picture of bend angles in mind, let’s look at what they mean in practical terms, specifically, as they relate to gain measurements.

When we measure gain, we typically aim the light source directly at the center of the screen and do the same with the light meter.  The essential reason for this is not because the center of the screen is special or that other considerations necessarily need to be made based on the center of the screen.  Rather, it is to control as many variables as possible thus simplifying the measuring process.

When the light is directly in front of the center of the screen, the rays that are incident to the screen’s center will measure 0° perpendicular to the surface and the reflections will be the same.  This means that by keeping the light source stationary and the meter trained on the screen center while moving it in 5° increments in an arc around the center, we can be as sure as possible that each measurement is taken at the intended bend angle.  If we put the light somewhere else and aimed the meter at a point other than the center, we would need to do a few extra calculations to determine at what angle the gain is being read.  It would, however, be possible to measure this way.

Incidentally, this is a handy lesson in why bend angles are so important.  That gain chart showing the relative brightness of the screen from all of those different angles is going to tell you roughly how bright the screen will appear at any point, not just the center we usually measure, so long as the bend angle can be determined for those points.

In almost all cases, the brightest area of the screen is going to be at 0° from which point the brightness will descend at some rate as the angles increase.  The more rapidly the gain falls the more likely it is that a hot spot will be visible.  The general rule states that the hot spot is most likely to appear inside the half gain angles, that is, the angles where the gain drops to half of what it was at the highest point.

What does it mean to be “inside the half gain angles”, exactly?  The angles we are interested in are, naturally, the bend angles.  In order to find them, we need to take the location of each viewer into account relative to the surface of the screen and we also need to know where the projector is positioned.  Assuming a projector mounted directly in front of the very topmost edge of the screen and a viewer seated in front of the bottom edge at the center, let’s look at what we might expect that viewer to see.

First, we need to know the smallest angle visible to him relative to the projector.  This will be the one ray that goes from the projector to the screen and directly into the eye of the viewer, giving us a 0° bend angle.  From some seating positions, this ray will not be visible but for this particular example, we can be fairly sure that the ray being reflected at about the center of the screen will give us the bend angle we’re looking for.  The angle of incidence at that spot will, as always, be equal and opposite to the angle of reflection.  Given that the projector and the viewer are on opposite edges of the screen, the angle from the projector to the screen’s center will match the angle from the center to the viewer’s eyes.

If the smallest angle is where the reflected ray and the viewer’s sightlines converge, the largest angles will be where the ray and the sightline are most divergent.  Generally speaking, these will be the points located at the corners of the screen.  Different projector and viewer positions will yield somewhat different solutions than this, but again, this example will have the largest bend angles at the corners.  Without entering into the specific trigonometry involved, let’s say that the bend angles at the corners will be a fairly reasonable 40°. 

When the largest and smallest bend angles are known, we can turn to the gain chart for the intended screen, such as the one in Figure 2.  The smallest angle in our example was 0° which means that our peak gain is 2.30, according to the chart.  The largest angle might be somewhere in the 40° area.  This gives us the lowest gain of 0.41.  Since the lowest gain is less than half of the peak gain, the chances of there being a hot spot are quite good.

This is not a guarantee, however.  It is possible to use a dimmer projector in order to minimize the visibility of a hot spot in situations where one is likely to occur.  Occasionally, someone may attempt to specify a bright projector with a high gain screen in hopes of creating a singularly bright display.  The result is usually a blinding hot spot that can be alleviated by simply reducing the light output of the projector. 

Another option is to increase the throw distance of the projector.  Compare the arrangements in Figure 1a and Figure 1b to see how moving the projector closer to the screen can increase the maximum bend angles while setting it farther back will reduce them.  If we used a long throw lens to turn the 40° bend angles at the corners of the screen in our previous example to less than 25°, the resulting minimum gain might not drop below half of the peak.  In this case, a hot spot might be avoided without making any other changes to the system.

A further alternative is to use screen materials that reflect light a little differently.  This brings us to the second important yet often overlooked component of uniformity issues: reflectivity.  Retro reflective materials reflect light primarily back towards the source instead of at an angle away from it.  Figure 3
uses a similar arrangement of projector, screen and viewer as Figure 1a but introduces a retro reflective surface.  As you can see, the bend angles become notably more acute as the rays trace back towards the lens instead of away from it. 

In other words, the bend angles at even the corners of the screen will not be nearly as wide as with the screen in the previous example because the reflected rays will not be nearly as divergent from the sightlines of the viewer. 

With these relatively narrow angles we can safely increase the gain of the screen quite a bit without introducing the same uniformity issues that would make an angular-reflective screen hot spot unrepentantly.  As you might suspect, this is exactly what we do. 

Figure 4 is a 3D representation of a gain chart for a retro reflective surface with a 2.8 gain.  The chart is presented in this fashion to accentuate the point that the bend angles apply in all directions and not just the two dimensions often illustrated.

As you may gather from the mountainous curve, the half angle on this screen is relatively narrow and again, with an angular-reflective screen, this would mean the same hot spot likelihood as in Figure 2. Because it is retro reflective, however, acceptable levels of brightness uniformity can be observed from a variety of seating positions, especially those that are as close to the projector as possible, thanks to the relatively small bend angles afforded by its reflective properties. 

A gain chart for this screen is, therefore, somewhat misleading.  It is not inaccurate and does not somehow result from faulty measuring techniques but it is important to understand that the chart is meant to be used with bend angles and not simply the angle between the viewer and the center or edges of the screen.  Furthermore, bend angles will be calculated differently for retro reflective screens because the light is reflected differently.  Instead of the angle of reflection being equal and opposite to the angle of incidence, both angles will be exactly the same.

As an aside, it is possible to see a hot spot on a retro reflective screen even despite the considerable improvement in brightness uniformity that it allows.  Granted, the absolute worst seating positions are somewhat impractical for normal viewing and the shadow cast by the viewer’s head will eclipse the bright center of the hot spot anyway.  More subtle uniformity differences may be seen from otherwise reasonable seats, however, so it is wise to remember that the same rules apply as before with the maximum and minimum simultaneously visible bend angles determining the peak and lowest gain for each viewer. 

The surest ways to avoid hot spots, in general, are to position the projector as far back from the screen as possible, to use a projector that is not too bright for its intended use and to use a material whose gain does not drop off too quickly or is retro reflective.  A somewhat more detailed model may be derived by more sophisticated methods utilizing some fairly tedious calculations.  Be sure to visit www.da-lite.com in the future for some computer aided illustrations of what these calculations can do.

-- Adam Teevan
   (ateevan@da-lite.com)

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