Bend loss

Bend loss is the extra attenuation of light in a fiber or waveguide that is curved. In a bend, the part of the guided mode on the outside of the curve has to travel faster than the part on the inside to keep its wavefront together, and beyond some distance from the axis it would have to outrun light in the cladding; the field there radiates away. The loss rises exponentially as the radius falls, so it is negligible above a certain radius and grows by orders of magnitude a little below it. Standard single-mode fiber (ITU-T G.652.D) is specified at no more than 0.1 dB for 100 turns of 30 mm radius at 1625 nm, bend-insensitive G.657.A1 fiber at no more than 0.75 dB for one turn of 10 mm radius at 1550 nm, and a 500 × 220 nm silicon waveguide, computed, loses 0.0003 dB to radiation in a 90° bend of 1.5 µm radius.

Why bends radiate

A bend can be described as a straight waveguide whose refractive index increases toward the outside of the curve, $n(x)(1+x\mathrm{/}R)$ for a bend of radius $R$ with $x$ measured outward from the axis (the conformal transformation of Heiblum and Harris). On the outside the tilted cladding index eventually exceeds the mode's effective index, at a distance of about $R(n_{\text{eff}}\mathrm{/}{n}_{\text{clad}}-1)$, and beyond that point the field is no longer evanescent but travels away. The mode reaches that point only through its evanescent tail, which decays exponentially, so the loss per unit length takes the form

$$\alpha (R)\approx \frac{A}{\sqrt{R}}\exp \text{\hspace{0.17em}⁣}(-\frac{R}{R_c}),R_c=\frac{3{\beta }^2}{2{\gamma }^3},$$

where $\beta$ is the propagation constant and $\gamma =\sqrt{{\beta }^2-k_0^2{n}_{\text{clad}}^2}$ the decay constant of the field in the cladding (Marcuse's curvature loss formula for step-index fiber). A weakly guided mode, with a small $\gamma$, has a large $R_c$ and must be bent gently. In a step-index model of standard single-mode fiber, $R_c$ is 0.9 mm at 1310 nm, 1.5 mm at 1550 nm and 1.75 mm at 1625 nm, so at 1550 nm the loss changes about tenfold for every 3.5 mm of radius. The bend also pushes the mode toward the outside of the curve, which adds a mismatch loss where a straight section joins a bent one.

Fiber

The bend limits of standard and bend-insensitive single-mode fiber are set in ITU-T recommendations G.652 and G.657:

G.657.A2 and B2 fibers are specified down to a 7.5 mm radius and B3 fibers down to 5 mm, using a smaller mode field diameter or a low-index trench around the core that holds the mode in place. Standard G.652 fiber has no limit at radii below 30 mm, and its loss there depends on the particular design; the measurement and the specifications are described in the macrobend loss entry.

Bend loss rises with wavelength, because the mode spreads further into the cladding and $\gamma$ falls as the V-number drops. The limits are therefore given at 1550 and 1625 nm, the longest wavelengths in use, and a bend that is harmless at 1310 nm can attenuate the L-band.

Macrobend loss in fiber is measured by wrapping it on mandrels of set radius and comparing the transmitted power with the straight fiber, as described in How to Measure Fiber Attenuation and Bend Loss.

Integrated waveguides

High-index-contrast waveguides confine the mode tightly and bend at radii of micrometers. Computed at 1550 nm for the TE-like mode of 500 nm wide silicon waveguides 220 nm thick in oxide, the radiation loss per 90° is:

In a strip at these radii the larger loss is the mismatch between the straight and the bent mode, 0.16 dB at each junction for a 1.5 µm bend and 0.09 dB for a 2 µm bend, which offset junctions or bends of gradually increasing curvature reduce. In a rib the light leaks sideways into the slab, so the thicker the slab, the larger the radius needed; the comparison is in strip vs rib waveguide. Radiation also limits the quality factor of small ring resonators: at a 1 µm radius the strip's radiation loss alone caps the Q at the order of ${10}^4$.

Macrobends and microbends

Macrobends come from routing and are controlled by keeping to a minimum radius. Microbends come from cable construction, coatings and pressure on the fiber, and accumulate over long lengths.

Bend loss as a sensor

Bend loss is the operating principle of one class of fiber-optic sensors: a fiber is held against a deformable structure whose movement changes the bend radius, and so the transmitted power. Strain, pressure and displacement sensors work this way, and longer wavelengths give a larger response, since the loss rises with wavelength.

Common questions

What causes bend loss?

In a bend, the field on the outside of the curve would have to travel faster than light in the cladding to keep pace with the rest of the mode, and beyond a certain distance from the axis it radiates instead. The loss grows exponentially as the radius shrinks.

What is the minimum bend radius of single-mode fiber?

Standard G.652 fiber is specified only at a 30 mm radius, which is the usual minimum for routing it. Bend-insensitive G.657.A1 fiber is specified down to 10 mm, A2 and B2 down to 7.5 mm and B3 down to 5 mm.

Why is bend loss higher at longer wavelengths?

Because the mode is less tightly confined at longer wavelengths: its field extends further into the cladding and reaches the radiating region more easily.

How tightly can a silicon waveguide bend?

A 500 × 220 nm silicon strip bends at a radius of 1.5 to 2 µm with negligible radiation loss; the limit then comes from the mismatch at the junctions and from fabrication. Ribs need 8 µm or more, depending on the slab thickness.

References: D. Marcuse, "Curvature loss formula for optical fibers," Journal of the Optical Society of America 66, 216 (1976). M. Heiblum and J. H. Harris, "Analysis of curved optical waveguides by conformal transformation," IEEE Journal of Quantum Electronics 11, 75 (1975). ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable, and G.657, Characteristics of a bending-loss insensitive single-mode optical fibre and cable. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991).