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For my project I use the end of a single mode fiber as a "transmitter". I need to set the divergence to 20 micro radians. Is there an equation how to calculate the divergence and the necessary optics to it?

Edit: I have two fiber ends at 100m distance from each other. Each should transmit light towards the other with 20 microradians divergence. I understand that there will be significant loss when coupling the received signal to the fiber. Once in the fiber, signal is separated with isolator. The question is how to specify the divergence for the output. I would like to have some equations since I want to use different wavelengths. At the moment I have 650nm and 1550nm fiber coupled lasers. (I checked the Thorlabs collimators, but they offer ~100 microradian divergence and the price is a little bit high for me. I would prefer to buy single lenses and adjust myself.)

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    $\begingroup$ can you describe your system in more detail? $\endgroup$
    – Bruce Lee
    Commented Feb 8, 2016 at 9:10
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    $\begingroup$ Looks just like a need for a single collimation lens, or even better, a quality photographic objective. If you need a such a tiny divergence, play for a while with (de)focusing. $\endgroup$
    – dominecf
    Commented Feb 8, 2016 at 9:14
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    $\begingroup$ 4 arcseconds? What's the wavelength? For green light that would require a lens of at least 1" diameter (telescope resolution is approx. 4.5arcsecInch/diamter). Your single mode fiber will have a numerical aperture of between 0.1-0.5 and that's the NA for your lens, too, if you want to collect most of the light. In practice you need to do a little better than that, of course, so I would go with a 1.5-2" (40mm?) lens. If you are thinking about near IR, you will have to roughly double the lens diameter. $\endgroup$
    – CuriousOne
    Commented Feb 8, 2016 at 9:23
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    $\begingroup$ To add to CuriousOne's comment: don't expect to collect all the light exiting the fiber. Figure out how much light you need in your output $20\mu radian$ beam and adjust accordingly. $\endgroup$
    – Carl Witthoft
    Commented Feb 8, 2016 at 12:32
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    $\begingroup$ take a look at fiber collimators like this $\endgroup$
    – scrx2
    Commented Feb 8, 2016 at 19:59

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THere's a relationship between the mode radius (d), divergence (theta) and wavelength. For single-mode fiber, d x theta ~ wavelength. So, if you want a narrower diverging beam you'll have to deal with a larger mode diameter in the same proportion.

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