Using a concave lens to increase divergence - picking the right strength
I've been looking into installing a lens in my projector to increase the divergence enough to make audience scanning safe. I've seen recommendations to use -3 diopter lenses in small rooms, but I wanted to be able to calculate appropriate strengths for different conditions.
Lens strength is often specified in diopters. It is simply the inverse of the focal length, that is:
D = 1/f
Where D is the strength in diopters (m^-1) and f the focal length in meters. A collimated beam that passes through a lens will get a half-angle divergence of:
θ = r0/|f| = r0*|D|
Where r0 is the radius of the collimated beam. Note that it's the absolute value of the strength that is used. A concave lens will give the same divergence as a convex of the same power. The convex lens will however have its focal point in front of the lens, which is not desirable. It is also interesting to note that the divergence is directly proportional to the beam diameter, which means that the initial size of the beams is very important and that they must be matched.
The size (radius) of the beam at a distance z is given by:
r = r0*z/|f| + r0 = r0*z*|D| + r0
For example, using a -3 diopter lens to diverge a beam with a 3 mm diameter and measuring at 4 meters:
r = 1.5 mm * 4 m * |-3| m^-1 + 1.5 mm = 19.5 mm
Which means that the beam will be 39 mm wide at 4 m. Assuming a circular beam with a flat profile (not very realistic though) the following equation can be used to calculate the irradiance, which is really what matters from a safety perspective:
E = Φ/A = Φ/(π*r²) = Φ/(π*r0²*z²*D²)
Where E is the irradiance (W/mm²) and Φ is the radiant flux (W). To convert to W/cm², multiply by 100. While the obtained values here should be taken with a grain of salt, because beam profiles are never flat, it is useful to see that doubling the strength of the lens (or distance) gives 4 times lower irradiance. The actual irradiance must be measured to assure safety.
Hope this will be useful to someone.
Now, while on the topic of lenses. Where the hell do I get one? I've visited all my local opticians and it turns out that they don't actually have these lenses. Well, except for one... but he wanted 120 € for a single uncut plastic lens
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Originally Posted by mixedgas
