# Nonlinear Length

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: the propagation distance in a medium over which nonlinear effects become substantial

The nonlinear length of a transparent medium is defined as the propagation distance over which nonlinear effects become substantial. Specifically, this is done in the context of self-phase modulation or cross-phase modulation in optical fibers, which is caused by an increase of the refractive index by the applied optical intensity. Here, one can define a nonlinear coefficient

$$\gamma = \frac{{2\pi }}{\lambda }\frac{{{n_2}}}{{{A_{{\rm{eff}}}}}}$$(with <$\lambda$> being the vacuum wavelength and <$A_\textrm{eff}$> the effective mode area) such that the phase shift occurring in self-phase modulation can be written as:

$$\varphi_\textrm{nl} = \gamma \: P \: L$$where <$P$> is the optical power and <$L$> is the propagation length. The nonlinear length is then

$${L_{{\rm{nl}}}} = \frac{1}{{\gamma \;P}}$$which is the length over which a nonlinear phase shift of 1 rad is obtained. At that order of magnitude, spectral broadening of an originally unchirped optical pulse becomes substantial. For strong spectral broadening by SPM, the fiber length needs to be substantially longer than that.

Obviously, that nonlinear length depends on the applied optical power and not only on the properties of the nonlinear medium itself.

## Effective Length

In the context of fiber nonlinearities, but also often encounters an *effective length* which is defined as

and has a substantially different meaning than the nonlinear length as defined above. It is *not* just an adaptation of the above formula for cases where the propagation loss (with an attenuation coefficient <$\alpha$>) has a substantial impact. Instead, it means the length of a hypothetical fiber with the same nonlinear coefficient but zero propagation losses, in which the same amount of nonlinear phase shift would be achieved.

One can easily see that the effective length converges to <$L$>, the actual length of the fiber, in the limit <$\alpha \rightarrow 0$>. The opposite limit is that of strong propagation losses, where only a negligible amount of the input power reaches the fiber end, and that results in <$L_\textrm{eff} = 1 / \alpha$>, independent of the actual fiber length.

That effective length is useful, for example, when comparing how strong nonlinear effects can be achieved with different fibers. For example, highly nonlinear fibers typically exhibit higher propagation losses than standard fibers, and thus a shorter nonlinear length, but at the same time they have a much increased nonlinear coefficient. The increase of nonlinearity may be higher than the decrease of the effective length, so that stronger nonlinear effects can be achieved.

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2021-07-13

Why do we have the vacuum wavelength in the first equation, although the nonlinear effect occurs in a medium having a linear refractive index, which we consider in the case of cross-phase modulation? Shouldn't we include <$n$> in the numerator of the nonlinear coefficient?

The author's answer:

The first part of the equation (<$2\pi / \lambda$>) gives you the linear phase change per unit length in vacuum (i.e., for <$n$> = 1). The product of nonlinear index <$n_2$> and optical intensity gives you the nonlinear change of refractive index (by definition of the nonlinear index). Therefore, the combination gives you the nonlinear phase change. The equation is correct as it is.

For cross-phase modulation, we also do not have a linear refractive index in the equation for the phase change.