Optical fibers can have different transverse refractive index profiles. Apart from such fibers where light is guided at the air–glass interface, the simplest index profile is a rectangular one, where the refractive index is constant within the fiber core, and is higher than in the cladding. Fibers of that kind are called step-index fibers. That term also includes designs with multiple index steps – for example, with additional rings of increased or depressed index.
The propagation modes of step-index fibers can be described with functions belonging to the family of Bessel functions, multiplied by an exponential phase factor exp(i β z) for the longitudinal phase variation, where β is a propagation constant (or phase constant). Concerning the radial dependence, the field strength in the core is proportional to the zero-order Bessel function of the first kind, and the cladding part is given by a modified Bessel function of the second kind. The mode function and its first derivative are always continuous at the core–cladding interface.
Various fiber parameters, in particular the numerical aperture and the V number, are originally defined only for step-index fibers, even though effective values are sometimes used for other fiber types.
For large V values, the number of modes is proportional to V2. For example, when the core area is scaled up while the numerical aperture is held constant, the number of modes is approximately proportional to the core area.
Deviations for Real Fibers
Multimode fibers often have a refractive index profile which is close to a perfect step-index profile. However, standard fabrication techniques for single-mode fibers often lead to significant deviations from this simple situation. In particular, preferential evaporation of the dopant during the collapse of the preform (assuming that the preform is made with inside chemical deposition) often leads to a pronounced dip of the refractive index profile at the center. Also, the index step can be somewhat smooth – more precisely described with a supergaussian function – due to diffusion during the fiber drawing process.
In some cases, deviations from a step-index profile are intentionally used in order to achieve certain guiding properties. For example, a region with depressed refractive index between core and cladding can introduce an additional cut-off wavelength, above which the propagation losses become very high.
Numerical Problems with Step-index Profiles
Although the step-index profile is mathematically very simple, it can be somewhat problematic in numerical simulations of beam propagation. Much lower numerical errors may be achieved e.g. by replacing a step-index profile with a supergaussian profile of high order, looking quite similar to the ideal rectangular profile. The index transition should be smoothed just such that it is sampled with a few numerical grid points. In that way, one may accurately simulate a fiber with nearly the same mode structure is a true step-index fiber. Anyway, real fibers usually also exhibit some smoothing of the core–cladding interface, caused by diffusion in the fiber drawing process before the fiber is cooled down.
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This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics AG. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!