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Thread: 2 equations for divergence

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    Default 2 equations for divergence

    Hi laserists,
    I'm making a wee app for doing various laser related calculations on the fly and i came across two different equations for beam divergence.
    On wikipedia it gives this equation(https://en.wikipedia.org/wiki/Beam_divergence):

    Θ = 2 x arctan(Δdiameter / 2 x length)

    ...whereas the common equation found here(http://iiith.vlab.co.in/?sub=1&brch=189&sim=342&cnt=1) is:

    Θ = Δdiameter / length

    All the online calculators seem to use the bottom equation so what is the top one about???
    cheers,
    Will

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    Quote Originally Posted by Discoordination View Post
    Hi laserists,
    I'm making a wee app for doing various laser related calculations on the fly and i came across two different equations for beam divergence.
    On wikipedia it gives this equation(https://en.wikipedia.org/wiki/Beam_divergence):

    Θ = 2 x arctan(Δdiameter / 2 x length)

    ...whereas the common equation found here(http://iiith.vlab.co.in/?sub=1&brch=189&sim=342&cnt=1) is:

    Θ = Δdiameter / length

    All the online calculators seem to use the bottom equation so what is the top one about???
    cheers,
    Will
    Both of those are real simple approximations. When I get on my lunch break I'll find you something valid for Gaussian beams,

    Steve
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    Qui habet Christos, habet Vitam!
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    I'm not sure i'm after the gaussian spot size as i'm guessing i'll just guesstimate the diameter in each dimension and aim to hit the same spot at each distance. Not got the tools to measure at aperture or the focal point of the lens.... You think it'd do any harm to average the 2 methods of calculation or is one inherently a more correct guestimate?

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    Every laser data sheet I've ever read has the initial spot size and the divergence. Usually in milliradians, ie starting with a 1 mm beam at 1 mil means a 1 meter spot at 1000 meters.
    `
    If you have the initial spot size, you use the formal Gaussian equation. This is further complicated by diode lasers usually having two divergences, as their beams are elliptical.
    `
    If you have no other means of doing it, the arctangent based equation is the one to go with,
    `
    Most hobbyists measure the beam divergence by taking an initial diameter measurement, then moving downrange and taking a second one. If you take it at the 10% and 90% power points, ie just inside the beam where the Gaussian function really starts to increase in intensity, its not a bad way, although pros will take the data at the 86% points. This works as long as you take the two diameters past the beam "waist" which is often near the exit point of the beam from the laser.
    `
    You then take the two diameters and the distance and work out the angle.
    `
    ie Theta is D2-D1/Z2-Z1 where D is the diameter and z is the distance.
    `
    see:
    `
    http://vlab.amrita.edu/?sub=1&brch=189&sim=342&cnt=1

    ~
    Steve
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