Speed of light in straight vs. circular optical fiber

What is the comparison of speed of light in an optical fiber when it is stretched in a straight line vs. when it is bent in a circle? Please provide computation or reference.

I think the bent fiber transmit light slower. In a straight fiber, the light is allowed to travel in a straight line together with other zig-zag paths. In a bent fiber, the straightest line it can travel is constrained by the inner and out radii of the curvature of the bend. The ratio of distance traveled or time taken on straight over circular path is $q=\frac1{2\sqrt d}\arccos\frac{1-d}{1+d}$ where $d$ is the ratio of the radius of the core fiber over the radius of the curvature of the central axis. For $d\in(0,1]$, $q<1$ because $$a)\quad \tan\sqrt d-\sqrt d=\int_0^\sqrt d \big((\sec x)^2-1\big)dx>0 \Longleftrightarrow\frac1{\sqrt{1+d}}>\cos\sqrt d\Longleftrightarrow\frac{1-d}{1+d}>2(\cos\sqrt d)^2-1=\cos2\sqrt d,$$ or $$b)\quad f(d):=\arccos\frac{1-d}{1+d}-2\sqrt d=\int_0^d f'(x)dx =-\int_0^d\frac{\sqrt x}{1+x}dx<0,$$ The shortest path in the circular fiber is longer than that in the straight one.

But this is a rough estimate, as I assume the straight line fiber is equivalent to having the light traveling in the center line in the bent fiber, and I am assuming the wavelength of the light is much smaller than the core radius of the fiber so that the light can be treated as a ray. A more rigorous analysis would give a more accurate answer. Surely the fully rigorous one would call for Maxwell wave equations in a circular waveguide. Is there a simple approximation in between that will give a good result?

• Just curious: why do you expect it to differ? – Asher Apr 16 '17 at 22:05
• @Asher: The (total internal) reflection paths and therefore the distance of the light traveled in the circular fiber are different from the straight fiber. – Hans Apr 16 '17 at 22:54
• do you mean cross-section of the fiber? then it is not "straight" but "rectangular" – aaaaaa Apr 17 '17 at 1:51
• Typical optical fiber can't be bent into a diameter smaller than about 4 cm without damage. That gives a ratio of over 4 million between the core diameter and the bend diameter. Do you still expect it to affect the propagation velocity significantly? – The Photon Apr 17 '17 at 2:23
• 1. That's probably less than the effect of the stress induced change in the refractive index. 2. You can't really use ray optics to analyze this situation, you have to use wave optics. – The Photon Apr 17 '17 at 4:41