# Step-index Fibers

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: optical fibers with a step-index refractive index profile

More general term: optical fibers

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 amplitude profiles of the propagation modes for any radially symmetric index profiles can be described as products of several factors:

$$A_{lm}(r, \varphi) = F_{lm}(r) \: e^{i l \varphi} \: e^{i \beta_{lm} z}$$containing the LP mode indices <$l$> and <$m$>. Specifically for step-index fibers, the radial functions <$F_{lm}(r)$> can be described with Bessel functions <$J_l$> (in the core) and <$K_l$> (in the cladding):

$$F_{lm}(r) = \begin{cases} a \: J_l(\beta_{\rm t} r) & \text{if } r \leq r_{\rm core}\\ b \: K_l(w r) & \text{if } r \geq r_{\rm core} \end{cases}$$where the constants <$a$> and <$b$> must be such that the solution is continuous. Those as well as <$\beta_{\rm t}$> and <$w$> depend on the mode indices. The radial derivative of <$F_{lm}(r)$> must also be continuous. The two continuity conditions can be simultaneously fulfilled only for specific eigenvalues <$\beta_{lm}$> for a given optical vacuum wavelength. These eigenvalues can be obtained with numerical methods. Note that developing a reliable, robust and efficient numerical algorithm for that, which works well in a wide range of cases (including very high mode indices), has to overcome various considerable difficulties.

The fiber of this example supports four modes, disregarding different orientations of modes with non-zero <$l$> and different polarization states.

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 <$V^2$>. 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.

## Case Study: Mode Structure of a Multimode Fiber

We explore various properties of guided modes of multimode fibers. We also test how the mode structure of such a fiber reacts to certain changes of the index profile, e.g. to smoothening of the index step.

## 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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