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Thread: Using a concave lens to increase divergence - picking the right strength

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    Default 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
    Last edited by tocket; 02-15-2010 at 00:38. Reason: Equation oops... :P

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    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 [/QUOTE]

    I guess in Europe you do not have kiosks selling replacement glasses without persciptions for 5-6$ a pair at every chemists??? (note for Americans, Chemist = Pharmacy Europe, I'm told...)

    If not, surplusshed.com

    Steve
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    Hope this will be useful to someone.

    THANK YOU!
    This thread should be a sticky.... Surpirisingly this math is left out of most optics texts..

    Steve
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    Quote Originally Posted by mixedgas View Post
    I guess in Europe you do not have kiosks selling replacement glasses without persciptions for 5-6$ a pair at every chemists??? (note for Americans, Chemist = Pharmacy Europe, I'm told...)

    If not, surplusshed.com

    Steve
    Not in Sweden. You can buy reading glasses (positive strength) in many places, but if you're nearsighted you have to go to the optician and pay $$$. There actually seems to be a law regulating this.

    Anyway, thanks to a certain forum member I am now sorted with the lenses.

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    Default lenses

    THorlabs or edmonds. They have quite a selection, and much of it can be low cost. Get an AR coated lens to match your lasers, and you are good to go!

    Quote Originally Posted by tocket View Post
    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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    Quote Originally Posted by tocket View Post
    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:

    Is that a mistake in the last line or have I missed a step. Its seemed to jump from 19.5mm to 39mm.

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    Radius vs diameter
    Laser people use diameter, scientists use radius

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    I can get you a -2.9 plano concave lens which will fit in front of your aperture.

    10 euro ex shipping?

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    Those calculations don't take into account the divergence of the beam before it enters the lens. If i set the lens strength to 0, with a 7mm beam in I get a 7mm diverged beam at a distance.

    However just say that, your original beam had a half angle divergence of 1mRad, and the lens added an additional 1mRad, would the resulting divergence be a simple addition of these two ?
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    Quote Originally Posted by krazer View Post
    Radius vs diameter
    Laser people use diameter, scientists use radius
    OK thanks, never noticed the difference in the formula. What comes of being non scientific (well for the last 30 years anyway!)

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