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What is Effective Length and Effective Area ? (concepts for understanding nonlinear effect in optical fibers)

>> Effective Length Le

The nonlinear interactions in optical fibers depends on the transmission length and the cross-sectional area of the fiber. The longer the link length, the more the interaction and the worse the effect of nonlinearity. However, as the signal propagates along the link, its power decreases because of fiber attenuation. Thus, most of the nonlinear effects occur early in the fiber span and diminish as the signal propagates.

Modeling this effect can be quite complicated, but in practice, a simple model that assumes that the power is constant over a certain effective length Le has proved to be quite sufficient in understanding the effect of nonlinearities.

Suppose Po denotes the power transmitted into the fiber and P(z) = Poe-αz denotes the power at distance z along the link, with α being the fiber attenuation. Let L denote the actual link length. Then the effective length is defined as the length Le such that

This yields

Typically, α = 0.22 dB/km at 1.55 μm wavelength, and for long links where L >> 1/α, we have Le ≈ 20 km.

Let’s look at the figure below for the effective transmission length calculation. In the figure, (a) is a typical distribution of the power along the length L of a link. The peak power is Po. (b) is a hypothetical uniform distribution of the power along a link up to the effective length Le. This length Le is chosen such that the area under the curve in (a) is equal to the area of the rectangle in (b).


>> Effective Area Ae

In addition to the link length, the effect of a nonlinearity also grows with the optical power intensity in the fiber.For a given power, the intensity is inversely proportional to the area of the core. Since the power is not uniformly distributed within the cross section of the fiber, it is convenient to use an effective cross-sectional area Ae, related to the actual area A and the cross-sectional distribution of the fundamental mode F(r,θ), as

where r and θ denote the polar coordinates.

The effective area, as defined above, has the significance that the dependence of most nonlinear effects can be expressed in terms of the effective area for the fundamental mode propagating in the given type of fiber.

For example, the effective intensity of the pulse can be taken to be Ie = P/Ae, where P is the pulse power, in order to calculate the impact of certain nonlinear effects such as Self-Phase Modulation (SPM).

The effective area of standard single mode fiber (SMF) is around  85 μm2 and that of Dispersion-Shifted Fiber (DSF) around 50 μm2. The dispersion compensating fibers have even smaller effective areas and hence exhibit higher nonlinearities.

Let’s look at the following figure, it shows the effective cross-sectional area. (a) shows a typical distribution of the signal intensity along the radius of optical fiber. (b) shows a hypothetical intensity distribution, equivalent to that in (a) for many purposes, showing an intensity distribution that is nonzero only for an area Ae around the center of the fiber.

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