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I am trying to focus some lasers into a single mode fiber optic cable which has a lens (ca. 0.7mm as far as I know) at the front on one end and a straight tip fiber optic connector on the other side (which then connects to a microscope). I am fairly certain that I am hitting the lens on the front of the cable with at least some of my laser light, but I am not seeing any light come out at the other end of the cable (ST connector). I think I should be, but I haven't worked with fiber optics before so I want to make sure that is actually the case.

Also, this setup I am using is apparently copied from another lab exactly, but it doesn't have any lenses besides the one in the cable and the light path for some of the lasers is over 1 m. This makes the laser spot larger than the entrance to the fiber, which would obviously reduce coupling efficiency. Might there be a reason there is no lens here, or would that be a simple way to improve the efficiency.

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Here is a video I found while writing this answer which probably does a better job than I do here. Video

It takes a bit of care to align a single mode optical fiber. I'll call the lens you refer to the "collimator". For light to couple into the fiber it is important that at the location of the fiber tip:

1) The mode of light is comparable to the mode field diameter in the fiber (typically few $\mu m$ for visible/IR wavelengths I am familiar with) You may need to do beam shaping upstream of the fiber collimator to ensure this. If the beam is truly visibly much bigger than the fiber tip then it is possible the coupling efficiency is VERY low and I would not be surprised you do not see light on the other end.

2) Once the mode size of light is comparable to the mode field diameter of the fiber it is of course necessary that the optical spot is spatially overlapped with the fiber core

3) It is also critical that light is approaching the fiber core at the appropriate angles.

All of this is typically accomplished by using mirrors and lenses upstream to control the beam size, position, and angle at the location of the fiber tip. The best trick to initially see light coming through the fiber is to use a fiber checker such as this one. You use this by plugging it into the downstream side of the fiber. It then sends light backwards out the fiber port you are trying to couple light into. You then overlap your incident beam with the back propagating fiber checker beam at multiple positions along the beam path (typically one close to the fiber collimator and one as far away as possible, this ensures matching of both position and angle.) This will get the position and angle right. If the wavelength of the fiber checker is pretty close to the wavelength of your incident beam then you can also match up the sizes of the two beams to ensure optimal coupling. If not you may have do some finagling or adjustment of the incident beam size to get optimal coupling.

After overlapping using fiber checker you can check if you see light on the other side of the fiber from your incident light. If you do then it is now best to optimize the efficiency by putting this light onto a power meter and tweaking any adjustable mirrors and lenses you have on the input side. This can be done using the "beam walking" technique.

Depending on your coupling efficiency requirements you may have to do a lot more or less work. Depending on beam quality and the ability to match the mode size it should be possible to get 70-80% coupling efficiency through a single mode fiber. 50% shouldn't be too difficult.

Some of what I say here may not be transferable to your particular situation depending on wavelengths, mode shapes, collimators etc.

edit: One more note I guess. Supposing you don't have a fiber checker it may be possible to do all of this "by eye" but depending on how much power you have it may be easier or harder. I think you certainly need at least a power meter on the output so that even at very very low coupling efficiency you can still see SOME signal of light coming through. You can then optimize on this to tweak it up to much higher efficiency. 80% of the battle is getting the first signal!

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