# Why is fiber coupling efficiency dependent on core diameter?

I read on Thorlabs' website that the coupling efficiency of a fiber is primarily dependent on the core diameter and numerical aperture of the fiber. It makes sense to me that if the beam of light going into the fiber has too great of an angle, the outer parts of the beam will fail to couple into the core of the fiber - which is why we need to make sure that the NA of the beam is less than or equal to that of the fiber. I don't understand though why core diameter affects coupling efficiency. It seems to me that fibers with different core diameters but equal NA will funnel the same amount of light because the NA angle is measured from the axis of the fiber so the allowed cone of light should be the same regardless of core diameter. Why exactly does core diameter affect fiber coupling efficiency?

On a related note, I've also read elsewhere that a larger diameter fiber with a lower NA has lower coupling capability than a smaller diameter fiber with a higher NA. Can you explain how that's possible?

Thanks!

The fiber diameter x NA = a constant for any given fiber type. Fundamentally, if a fiber is a single-mode fiber (step index) then the mode-field diameter x far-field divergence angle (related to NA) is the smallest possible product.

To compute the coupling efficiency between a free-space distribution and then mode of the fiber, one computes what's known as an overlap integral (from a reciprocity theorem on E&M). This integral takes the, assuming a single-mode fiber, form Integrate[ Exp(-(x/w)^2 F(x) dx], where w is the mode-radius of the fiber and F(x) is whatever the free-space mode is. The extension to 2-D is trivial.

So, where am I going with this? From the integral it's apparent that the mode coupling efficiency depends upon the mode diameter of the fiber. Intuitively this makes sense if you think about certain limiting cases: for a mode-size that is very small,if the free-space mode is much larger than only a fraction of the total power will be subtended by the mode of the fiber, resulting in less than ideal coupling efficiency.

The main physical law is the law of total reflection. The main task is to keep the beam within the fiber.

Total reflection only happens at the border from an optical more dense to a less dense medium and below a certain angle.

When we now bend the fiber we will increase the angle, because the beam travels straight.

By using a smaller fiber this angle stays small.