The idea is to form an image of the sun on the core at the tip of the fiber at an image size that best matches the mode of the fiber. The size of the image is determined by the focal length of the lens. The sun has an angular subtense of 9.3 mrad, so the size of the image will be 0.0093*focal length. Choose the focal length of the lens to match the mode size of your fiber. Typical step-index multi-mode fibers have a simple top hat mode shape that will match the image of the sun very well, so coupling can be very efficient. However, you will still have loss from the Fresnel reflection particularly if the fiber tip isn't anti-reflection coated. I wouldn't worry about that small inefficiency too much. You will still get plenty of light. Since you can't match the Gaussian mode shape of a single-mode fiber, coupling would be less efficient. I also don't think you would want to work with the very short focal length microscope objective you would need to make the mode that small. So, you probably want multi-mode fiber with a reasonable core size. Plastic fibers are available with multi-step cores. These also do not have a top hat mode shape and typically have a small mode size to match standard single mode fibers. Stay away from these as well. A simple step-index PMMA fiber with a large core is your best bet.

With the focal length determined, the diameter of the lens determines the NA (numerical aperture) which is the cone angle with which the light will enter the fiber. The NA of the lens is sin(atan(used diameter/(2*focal length)) I say "used diameter" because some of the total lens diameter will be lost to mounting. The NA of the lens does not have to be as large as the fiber. For a multimode fiber, as long as the core size is matched, there is no coupling efficiency penalty to not matching the NA. However, the larger the NA of the lens the more sunlight will be collected and injected. Plastic fiber typically has a very large NA and I don't think you are likely to be able to use all of it, much less go any higher. However, I will mention that if the lens did exceed the NA of the fiber, the extra light in the cone which is at a larger NA than the fiber can accept will be lost. It doesn't hurt anything to have extra. The extra light just won't propagate down the fiber.

The sun is not a point source, but it isn't a very large source (in angular subtense) and you can make a very good image with a plano-convex lens. The plano side goes towards the fiber. There are also nearly plano-convex lenses called "best form lenses" that will do even a little bit better. As you choose to use larger and larger NA to collect more light, the image from the outer edges of the larger lenses starts to not focus well at the fiber. At larger NA you might want to move up to an aspheric lens. Aspheric lenses which are plano on one side and meant for exactly this purpose are readily available at Edmund optics or thorlabs. They are available in a variety of focal lengths and NAs, and the molded or replicated versions can be reasonably cheap.

Another big problem is that the sun is broad band. Due to dispersion in the lens material, not all wavelengths will be well focused at the fiber. To fix this you can move up a little further and use an achromatic doublet or triplet. These too are available at thorlabs and edmund (and many many others) for surprisingly reasonable prices. These won't be available in quite as high NA as the aspheres, but a quick look shows that they do sell pretty high NA lenses.

So, as an example: part number TRH254-040-A-ML available for $104 from Thorlabs is a hastings triplet achromatic lens designed for infinite conjugates (exactly what we are trying to do here) with a 1" diameter and a 40 mm focal length. That will produce a 372 um image of the sun with an NA of 0.3. That will couple very well into a 400 um core PMMA fiber. It doesn't quite use the available 0.5 NA of the PMMA fiber, but it still stuffs in a lot of light. Alternatively, you could use an asphere to come closer to the NA and thus either couple more light into the same fiber, or couple all of the 1" aperture into a smaller fiber. However, you would lose some efficiency in some parts of the spectrum due to chromatic aberration.

One advantage of using as large a core fiber as you can (where usually bending radius is what drives towards smaller) is it is easier to align. Getting the focal spot centered on a 400 micron fiber is 4X easier than centering on a 100 um fiber. The larger spot size and longer focal length also has a longer depth of focus, so it is similarly easier to align in the focus dimension.

In order to optimize the alignment, you generally will need a precision three axis stage. This can be the most expensive part of the project. If you are handy, you might be able to fabricate something good enough on your own based on some fine thread screws. On the other hand, if you can afford it, you could just buy a three axis alignment stage.

Mike Albert Thanks for your super detailed reply!

use Gradient Index Micro Lens

for more info: https://www.newport.com/g/fiber-coupling-optics