Depth of Field
Projecting on a curved screen not only creates geometrical distortion, it also affects focus. Warping software like Nthusim can correct distortion, but when it comes to focus it’s pretty much “what you see is what you get”. The good news is that contemporary video projectors are much better at maintaining focus on curved screens than were the old CRT units. Nonetheless, if you’re planning on using a curved projection screen, it’s good to know ahead of time how image focus will be affected.
Depth of field for a projection system is a zone within which an acceptably focused image can be formed on a screen. For example, if we adjust a projector to have “perfect” focus and find we can move the screen eight inches closer to or away from the projector before the image becomes too blurry, the depth of field is 16 inches. When projecting on a curved screen, the screen must fit within the depth of field if we’re to have an acceptably focused image.
Depth of field is determined by the projection lens, the distance from the lens to the screen, and a definition of acceptable focus from a given viewpoint.
If we look at all possible light paths from a projection lens to a perfectly focused spot on a screen we have a cone of light with its base at the lens and its apex just touching the screen. If we move the screen closer to the lens or farther away (without tweaking the focus) the spot grows in size as the screen intercepts widening portions of the light cone. The spot becomes “unfocused”.
Focus and lack of focus are defined with respect to the visual acuity of the human eye. An image is out of focus when it appears to be out of focus, not when the focus deviates microscopically from a theoretical perfect image. In photography an image viewed at a one foot distance is regarded as being acceptably focused if a bright “point” within the image is smaller than a 0.01” disk. If the image is viewed at a greater distance, the allowable disk size is proportionately larger. By this standard, a projection screen positioned 10 feet from the viewpoint will appear to be acceptably focused (to the average human eye) if a “point” is no larger than a 0.1” disk.
Let’s calculate the depth of field of a realistic projection system. Suppose we have a projector mounted above and behind the viewpoint such that the throw distance is 12 feet. Let’s put the viewpoint eight feet from the screen.
The first step is to determine the dimensions of the light cone. The light cone base diameter isn’t the same as the diameter of the lens. Instead it equals the size of the lens exit aperture. The exit aperture isn’t normally listed in the projector specifications. Fortunately we can calculate it by dividing the projection lens focal length by the lens f-number, both of which usually are listed. Our projector has a lens with a 24mm focal length at an f-number of 2.55. This gives an exit aperture of 9.4mm or 0.37”. So, our light cone has a base diameter of 0.37” and a length of 144”.
The next step is to determine the acceptable disk size. Since the distance from the viewpoint to the screen is eight feet, we’ll use a largest acceptable size of 8 * 0.01” = 0.08”.
If we chop off the tip of the light cone at the front limit to the depth of field we will have a small cone with a base diameter equal to the acceptable disk size and a length equal to the front depth of field. This small cone has the same proportions as the original projection light cone. We can determine the front depth of field by noting that the ratio of the front depth of field to the disk diameter is the same as the ratio of the projection light cone length to its base diameter.
There is also a rear depth of field which in this analysis has the same length as the front depth of field. So, the total depth of field is 62.2”, twice what we calculated above.
Strictly speaking the front and rear depths of field are not equal. The acceptable focus disk size at the front of the depth of field range is closer to the viewpoint so should be smaller. At the rear of the range the disk is farther away so can be larger. The results are that the front depth of field is actually smaller than calculated while the rear depth of field is larger. These errors tend to cancel, but don’t do so exactly.
A second example shows how typical installation constraints can have a large effect on depth of field. Suppose we use the same projector in a smaller room. We’ve reduced the throw distance to six feet and moved the viewpoint to only six feet from the screen so the allowable spot size shrinks to 0.06”.
Look what happens to the front depth of field:
What might have been expected to be a small accommodation has had a profound effect on depth of field. Whether this makes a system non-viable depends on the size and curvature of the screen.
CRT projectors have lenses with large exit apertures, and as a result, have a quite limited depth of field. The large lens size is required to collect the light coming from the large diameter CRT faces. LCD and DLP projectors have focused light sources illuminating light modulators much smaller than the CRT faceplates. These projectors produce excellent images using lenses with smaller exit apertures, and we benefit from the relatively generous depth of field performance. Most LCD and DLP projectors can be expected to provide good focus on moderately curved screens. However, if you plan on using a particularly short throw, a deeply curved screen, or find the 0.01” disk at one foot too loose a standard, consider using this simple analysis to check your plans before committing yourself.
Mike Powell, author of
Building Recreational Flight Simulators and
Building Simulated Aircraft Instrumentation.